ff1588d761d55f48f61a52249e1dcc24

<!DOCTYPE html>

<title>Spurklasseoperator</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Spurklasseoperator</h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Spurklasse-Operatoren werden in der <a href="Mathematik" title="Mathematik">mathematischen</a> Disziplin der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a> untersucht. Sie bilden eine wichtige Klasse von linearen Operatoren auf unendlichdimensionalen Räumen. Bei ihnen bleiben im Gegensatz zu allgemeinen Operatoren einige Eigenschaften aus dem endlichdimensionalen Fall erhalten, das betrifft insbesondere ihre Darstellung als Summe eindimensionaler Operatoren.

In wichtigen Fällen überträgt sich der aus der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> bekannte Begriff der <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> auf diese Operatoren, was zu ihrem Namen geführt hat. In der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> treten die Spurklasseoperatoren als <a href="Dichtematrix" class="mw-redirect" title="Dichtematrix">Dichtematrix</a> auf.

<a href="Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> stieß bei seiner Untersuchung des <a href="Satz_vom_Kern" class="mw-redirect" title="Satz vom Kern">Satzes vom Kern</a> aus der <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">Distributionentheorie</a> ebenfalls auf Operatoren der hier behandelten Art und nannte sie nukleare Operatoren (lat. nucleus = Kern). Dies führte dann zum Begriff des <a href="Nuklearer_Raum" title="Nuklearer Raum">nuklearen Raums</a>.

Die Terminologie ist nicht einheitlich, manche Autoren definieren Spurklasse-Operatoren nur auf <a href="Hilbertraum" title="Hilbertraum">Hilberträumen</a> und nukleare Operatoren auf Banach-Räumen. Dieser Artikel behandelt die nuklearen Operatoren zunächst auf <a href="Hilbertraum" title="Hilbertraum">Hilberträumen</a>, dann allgemeiner auf <a href="Banachraum" title="Banachraum">Banachräumen</a> und schließlich auf <a href="Lokalkonvexer_Raum" title="Lokalkonvexer Raum">lokalkonvexen Räumen</a>.

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ein <a href="Vektorraum" title="Vektorraum">Vektorraum</a> über dem Körper der reellen oder komplexen Zahlen. Ein eindimensionaler Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:E\rightarrow E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:E\rightarrow E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa8b954785dbe275044b591e1a0a7627b00fb3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.846ex; height:2.176ex;" alt="{\displaystyle A:E\rightarrow E}" loading="lazy"> ist ein Operator der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x):=f(x)\cdot y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x):=f(x)\cdot y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c5814fce075f7318cde357fbcdba35bdf00b27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.88ex; height:2.843ex;" alt="{\displaystyle A(x):=f(x)\cdot y}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/734cccfb5c201c64235d85862699e0084cf5c806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.772ex; height:2.509ex;" alt="{\displaystyle y\in E}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in E'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in E'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7deaae0714d0bd8baa51e9d529997b04956dab6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle f\in E'}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a574600572696493d48300245a45b8de0638ce21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}" loading="lazy"> den <a href="Dualraum" title="Dualraum">Dualraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> bezeichnet. In der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>, d. h. im Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\mathbb {K} }^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E={\mathbb {K} }^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd691ba2987a159bdc3c6c6256ec197d7d08c1cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.901ex; height:2.343ex;" alt="{\displaystyle E={\mathbb {K} }^{n}}" loading="lazy">, kann jede lineare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:E\rightarrow E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:E\rightarrow E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa8b954785dbe275044b591e1a0a7627b00fb3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.846ex; height:2.176ex;" alt="{\displaystyle A:E\rightarrow E}" loading="lazy"> als <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{i,j})_{i,j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{i,j})_{i,j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee0f76738742dac775ee80bb27f6c5b98c3a2682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.908ex; height:3.009ex;" alt="{\displaystyle (a_{i,j})_{i,j}}" loading="lazy"> bzgl. einer <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\ldots e_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots e_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b9eadbb5efca7cbbfaedea5451da2b87a8bbb7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.584ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots e_{n}}" loading="lazy"> dargestellt werden.

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\mathbb {K} }^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in {\mathbb {K} }^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9b5ca89d8a1c58b9ce3f1b69c67d486fb4b7686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.197ex; height:2.343ex;" alt="{\displaystyle x\in {\mathbb {K} }^{n}}" loading="lazy"> gilt dann

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Ax=(a_{i,j})_{i,j}\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}=\sum _{i=1}^{n}(a_{i,1},\ldots ,a_{i,n})\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\cdot e_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/840edc1580b83ef5ccb0042e89d89761cbf2e531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:54.956ex; height:10.509ex;" alt="{\displaystyle Ax=(a_{i,j})_{i,j}\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}=\sum _{i=1}^{n}(a_{i,1},\ldots ,a_{i,n})\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\cdot e_{i}}" loading="lazy">.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ist also eine Summe eindimensionaler Operatoren. Um das auf unendlichdimensionale Räume übertragen zu können, muss man unendliche Summen eindimensionaler Operatoren bilden und daher Vorkehrungen für deren Konvergenz treffen. Das führt zu folgender Definition:

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> zwei normierte Vektorräume. Ein Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:E\rightarrow F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:E\rightarrow F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93ff6b7547b10ceab4a51f1c5e021a8d7c6a56a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.811ex; height:2.176ex;" alt="{\displaystyle A:E\rightarrow F}" loading="lazy"> heißt nuklear, falls es Folgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc87f936c2f7d77029ef0eb03494fc8533b123bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.476ex; height:2.843ex;" alt="{\displaystyle (a_{n})_{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a447cb7ffab7af3e68d8c04ee2a1c38059ef135f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle (f_{n})_{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\,'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22a1e4e62eb25cfead498cd4b55b305f64098ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.847ex; height:2.509ex;" alt="{\displaystyle E\,'}" loading="lazy"> gibt mit

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\|a_{n}\|\cdot \|f_{n}\|<\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fe525a63bf83b1fffefcf0999279fd102625e44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.299ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }\|a_{n}\|\cdot \|f_{n}\|<\infty }" loading="lazy"></dd></dl>

und

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/731517c4d7bbaf8aae819c044b6e7ddc7298b446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.668ex; height:6.843ex;" alt="{\displaystyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)a_{n}}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30b1971b01bc31d5b816f03cc7e1d9215d6c2ad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.946ex; height:2.176ex;" alt="{\displaystyle x\in E}" loading="lazy">. Eine solche Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> heißt eine nukleare Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. Diese ist jedoch nicht eindeutig.

Die nukleare Norm oder Spurnorm eines nuklearen Operators ist definiert als

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|A\|_{1}=\inf \sum _{n=1}^{\infty }\|a_{n}\|\|f_{n}\|,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81c597c59abeae9ad98eb515889b6a6c9e3c3dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.104ex; height:6.843ex;" alt="{\displaystyle \|A\|_{1}=\inf \sum _{n=1}^{\infty }\|a_{n}\|\|f_{n}\|,}" loading="lazy"></dd></dl>

wobei das Infimum über die Folgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc87f936c2f7d77029ef0eb03494fc8533b123bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.476ex; height:2.843ex;" alt="{\displaystyle (a_{n})_{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a447cb7ffab7af3e68d8c04ee2a1c38059ef135f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle (f_{n})_{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\,'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22a1e4e62eb25cfead498cd4b55b305f64098ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.847ex; height:2.509ex;" alt="{\displaystyle E\,'}" loading="lazy"> gebildet wird, welche eine nukleare Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ergeben.

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n}\in \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45004753c427fc1fddef5cd6425280b4ee93f510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.341ex; height:3.176ex;" alt="{\displaystyle (a_{n})_{n}\in \ell ^{1}}" loading="lazy"> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon \ell ^{p}\rightarrow \ell ^{p}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon \ell ^{p}\rightarrow \ell ^{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed7e4f6148bf5923abeb1dd0123a1c29d5618b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.449ex; height:2.343ex;" alt="{\displaystyle A\colon \ell ^{p}\rightarrow \ell ^{p}}" loading="lazy"> definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n}\mapsto (a_{n}x_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n}\mapsto (a_{n}x_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/764ce566151d5bab84129bc59a9a45fb7d7d6c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n}\mapsto (a_{n}x_{n})_{n}}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> nuklear mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|A\|_{1}\leq \sum _{n}|a_{n}|}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|A\|_{1}\leq \sum _{n}|a_{n}|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d4c35428d644f0a31fda651e3b8b8f777fda20f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.705ex; height:5.509ex;" alt="{\displaystyle \|A\|_{1}\leq \sum _{n}|a_{n}|}" loading="lazy">. Im Hilbertraumfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d62e4100b94c1939c67f2d4b8580d26c78106c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}" loading="lazy"> gilt Gleichheit.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\colon [0,1]\times [0,1]\rightarrow {\mathbb {C} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93786ff3a1129e63cf2fddc066105eccb92010aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.683ex; height:2.843ex;" alt="{\displaystyle k\colon [0,1]\times [0,1]\rightarrow {\mathbb {C} }}" loading="lazy"> stetig, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon L^{\infty }[0,1]\rightarrow L^{\infty }[0,1]}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon L^{\infty }[0,1]\rightarrow L^{\infty }[0,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/596f26952bd9f947d6f56f1e53e33f7bf050aba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.613ex; height:2.843ex;" alt="{\displaystyle A\colon L^{\infty }[0,1]\rightarrow L^{\infty }[0,1]}" loading="lazy"> sei definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Af)(t):=\int _{0}^{1}k(t,s)f(s)\mathrm {d} s}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Af)(t):=\int _{0}^{1}k(t,s)f(s)\mathrm {d} s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba8a623dae1655aee4cda75b507bf6a92a7c3d0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.677ex; height:6.176ex;" alt="{\displaystyle (Af)(t):=\int _{0}^{1}k(t,s)f(s)\mathrm {d} s}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> nuklear mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|A\|_{1}\leq \int _{0}^{1}\sup _{t}|k(t,s)|\mathrm {d} s}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|A\|_{1}\leq \int _{0}^{1}\sup _{t}|k(t,s)|\mathrm {d} s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88da9f4165ebed404d1eb32cc396e0326329e878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.676ex; height:6.176ex;" alt="{\displaystyle \|A\|_{1}\leq \int _{0}^{1}\sup _{t}|k(t,s)|\mathrm {d} s}" loading="lazy">.</li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon \ell ^{2}\rightarrow \ell ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20a6c02a7f70c97084fc1ffb2247784ac1a0297e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.439ex; height:2.676ex;" alt="{\displaystyle A\colon \ell ^{2}\rightarrow \ell ^{2}}" loading="lazy"> definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n}\mapsto ({\tfrac {1}{n}}x_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n}\mapsto ({\tfrac {1}{n}}x_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05a68e3d2d844882e943c5d08a2e5a62366bc441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.588ex; height:3.343ex;" alt="{\displaystyle (x_{n})_{n}\mapsto ({\tfrac {1}{n}}x_{n})_{n}}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ein <a href="Kompakter_Operator" title="Kompakter Operator">kompakter Operator</a>, der nicht nuklear ist.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einfache_Eigenschaften">Einfache Eigenschaften</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E,F)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E,F)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aeec51324941eaf53b0d74004a9b2d119799d89e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.423ex; height:2.843ex;" alt="{\displaystyle N(E,F)}" loading="lazy"> die Menge aller nuklearen Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\to F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84d9e3dcf392a3538226db6bda1b7e2dd73c32e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.13ex; height:2.176ex;" alt="{\displaystyle E\to F}" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> <a href="Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum">vollständig</a>, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E,F)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E,F)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aeec51324941eaf53b0d74004a9b2d119799d89e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.423ex; height:2.843ex;" alt="{\displaystyle N(E,F)}" loading="lazy"> mit der nuklearen Norm ein Banachraum. Die Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\to F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84d9e3dcf392a3538226db6bda1b7e2dd73c32e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.13ex; height:2.176ex;" alt="{\displaystyle E\to F}" loading="lazy"> mit endlichdimensionalem Bild liegen dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E,F)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E,F)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aeec51324941eaf53b0d74004a9b2d119799d89e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.423ex; height:2.843ex;" alt="{\displaystyle N(E,F)}" loading="lazy"> und jeder nukleare Operator ist <a href="Kompakter_Operator" title="Kompakter Operator">kompakt</a>.

Die nuklearen Operatoren haben die sogenannte Ideal-Eigenschaft: Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,\,F,\,G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E,\,F,\,G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3036a63f4afbbba7bdc8c8ccd76a3229636b93dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.185ex; height:2.509ex;" alt="{\displaystyle E,\,F,\,G}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> normierte Räume, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon E\to F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon E\to F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79acbbd701b514cbc1ad74026adf3d2bc5a7975b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.908ex; height:2.176ex;" alt="{\displaystyle A\colon E\to F}" loading="lazy"> sei nuklear und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\colon G\to E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\colon G\to E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f48ef4f271f20562b9febb2ff95ba270a17bf78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.014ex; height:2.176ex;" alt="{\displaystyle B\colon G\to E}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\colon F\to H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\colon F\to H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/842c60a6448460e93280edcfaede27d65549f476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.219ex; height:2.176ex;" alt="{\displaystyle C\colon F\to H}" loading="lazy"> seien stetige lineare Operatoren. Dann ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CAB\colon G\to H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle CAB\colon G\to H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d75579031fc69f45ec624d14e82a248f32e5593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.812ex; height:2.176ex;" alt="{\displaystyle CAB\colon G\to H}" loading="lazy"> nuklear und es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|CAB\|_{1}\leq \|C\|\|A\|_{1}\|B\|}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|CAB\|_{1}\leq \|C\|\|A\|_{1}\|B\|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c4a20f218c17602727a24202494d6c6e7c37a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.053ex; height:2.843ex;" alt="{\displaystyle \|CAB\|_{1}\leq \|C\|\|A\|_{1}\|B\|}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}" loading="lazy"> die <a href="Operatornorm" title="Operatornorm">Operatornorm</a> sei. Es gilt stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|\leq \|\cdot \|_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|\leq \|\cdot \|_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32a538e1adf21c821657631368496a4814d98fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.161ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|\leq \|\cdot \|_{1}}" loading="lazy">

Speziell ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E):=N(E,E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E):=N(E,E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fc5b856cb48f1f071b3f0673bb09b3916ca9a53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.852ex; height:2.843ex;" alt="{\displaystyle N(E):=N(E,E)}" loading="lazy"> ein <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Ideal</a> in der Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7efcbf46eeaf242b4a57e27413d69d8de0b20370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.843ex;" alt="{\displaystyle B(E)}" loading="lazy"> der stetigen linearen Operatoren auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbe797df2434f2e14ced1b8d3440dbd580799b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.649ex; height:2.843ex;" alt="{\displaystyle N(E)}" loading="lazy"> mit der nuklearen Norm ist eine <a href="Banachalgebra" title="Banachalgebra">Banachalgebra</a>, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ein <a href="Banachraum" title="Banachraum">Banachraum</a> ist.

<div class="mw-heading mw-heading2"><h2 id="Nukleare_Operatoren_auf_Hilberträumen">Nukleare Operatoren auf Hilberträumen</h2></div>

Im Hilbertraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=F=H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=F=H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf4db068f077608d2cd822a514090be922566b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.777ex; height:2.176ex;" alt="{\displaystyle E=F=H}" loading="lazy"> sind die Verhältnisse einfacher. In diesen Räumen sind die nuklearen Operatoren 1946 erstmals durch <a href="Robert_Schatten" title="Robert Schatten">Robert Schatten</a> und <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> untersucht worden.

Jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}\in H'}">

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}\in H'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4737212634c96990488c5b7358e1e6a534f709c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.987ex; height:2.843ex;" alt="{\displaystyle f_{n}\in H'}" loading="lazy"> ist nach dem <a href="Satz_von_Fr%C3%A9chet-Riesz" class="mw-redirect" title="Satz von Fréchet-Riesz">Satz von Fréchet-Riesz</a> von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}(x)=\langle x,y_{n}\rangle }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}(x)=\langle x,y_{n}\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bd41645136d6f5f5caa283c3de2af54ab5038e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.126ex; height:2.843ex;" alt="{\displaystyle f_{n}(x)=\langle x,y_{n}\rangle }" loading="lazy"> mit einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n}\in H}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{n}\in H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3e2fb16eed7b6bbc53864a671a19d7a2413687d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.262ex; height:2.509ex;" alt="{\displaystyle y_{n}\in H}" loading="lazy">. Eine nukleare Darstellung eines Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon H\rightarrow H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon H\rightarrow H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1d5b62b891706fc42b3b56e6c8481b646127f93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.518ex; height:2.176ex;" alt="{\displaystyle A\colon H\rightarrow H}" loading="lazy"> hat daher die Gestalt

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x)=\sum _{n=1}^{\infty }\langle x,y_{n}\rangle x_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d419d3d01c2f40cc5815583c617f711bfb77e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.414ex; height:6.843ex;" alt="{\displaystyle A(x)=\sum _{n=1}^{\infty }\langle x,y_{n}\rangle x_{n}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n},y_{n}\in H}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n},y_{n}\in H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6337cf74e500099a6a04ce3f3890d37bb4f68bc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.844ex; height:2.509ex;" alt="{\displaystyle x_{n},y_{n}\in H}" loading="lazy"> und

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|\|y_{n}\|<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c47619bf970919bea739f1d5469fda9b78219a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.367ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|\|y_{n}\|<\infty .}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{m})_{m}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (e_{m})_{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce69c2fc95426bec43fbc947e55f2a98e0ca4d51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.243ex; height:2.843ex;" alt="{\displaystyle (e_{m})_{m}}" loading="lazy"> eine beliebige <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">, so konvergiert für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fcddc7a2f7de54d513a1d4606eba2a9f03ac25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle A\in N(H)}" loading="lazy">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{m}\langle Ae_{m},e_{m}\rangle =\sum _{n}\langle x_{n},y_{n}\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/006bab6e8659cea74d0190d129fa6915df45b092.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.661ex; height:5.509ex;" alt="{\displaystyle \sum _{m}\langle Ae_{m},e_{m}\rangle =\sum _{n}\langle x_{n},y_{n}\rangle }" loading="lazy">,</dd></dl>

wobei die linke Summe als Limes des <a href="Netz_(Topologie)" title="Netz (Topologie)">Netzes</a> aller endlichen Teilsummen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> zu lesen ist (d. h. als <a href="Unbedingte_Konvergenz" class="mw-redirect" title="Unbedingte Konvergenz">unbedingte Konvergenz</a>). Diese Zahl ist daher unabhängig von der Wahl der Orthonormalbasis und auch unabhängig von der Wahl der nuklearen Darstellung, sie wird die Spur von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> genannt und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dd6a975367c32d4ebf279de22010257974fbbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.137ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (A)}" loading="lazy"> bezeichnet. Wegen des englischen Wortes trace für Spur findet man auch häufig die Bezeichnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aa14fce2744b8ee431a4946b2bf8c760df8212b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.369ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (A)}" loading="lazy">.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fcddc7a2f7de54d513a1d4606eba2a9f03ac25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle A\in N(H)}" loading="lazy"> <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungiert</a> und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda _{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8a9d843d952fd8553c559b3d9b28d28db711bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.601ex; height:2.843ex;" alt="{\displaystyle (\lambda _{n})_{n}}" loading="lazy"> die Folge der mit Vielfachheiten gezählten <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \|A\|_{1}=\sum _{n}|\lambda _{n}|}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \|A\|_{1}=\sum _{n}|\lambda _{n}|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8869d5446e7e474a2ae896387af4188a98911904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.147ex; height:3.009ex;" alt="{\displaystyle \textstyle \|A\|_{1}=\sum _{n}|\lambda _{n}|}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \operatorname {Sp} (A)=\sum _{n}\lambda _{n}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \operatorname {Sp} (A)=\sum _{n}\lambda _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af9aa09200bb0cfea85f4753ba11ec0c5ec9a43c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.869ex; height:3.009ex;" alt="{\displaystyle \textstyle \operatorname {Sp} (A)=\sum _{n}\lambda _{n}}" loading="lazy">. Für allgemeines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fcddc7a2f7de54d513a1d4606eba2a9f03ac25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle A\in N(H)}" loading="lazy"> ist die Eigenwertfolge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda _{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8a9d843d952fd8553c559b3d9b28d28db711bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.601ex; height:2.843ex;" alt="{\displaystyle (\lambda _{n})_{n}}" loading="lazy"> absolut summierbar und es ist

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n}|\lambda _{n}|\leq \|A\|_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18dfb54cd821e2ebb1828bc6728eb4ebf6122db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.147ex; height:3.009ex;" alt="{\displaystyle \textstyle \sum _{n}|\lambda _{n}|\leq \|A\|_{1}}" loading="lazy">.

Als weitere Charakterisierung kann man zeigen, dass ein Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c880b5d87ed10fa5b19e2bd9e3e77194f9dab803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.221ex; height:2.843ex;" alt="{\displaystyle A\in B(H)}" loading="lazy"> genau dann nuklear ist, wenn er das Produkt zweier <a href="Hilbert-Schmidt-Operator" title="Hilbert-Schmidt-Operator">Hilbert-Schmidt-Operatoren</a> ist.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> spielt eine zentrale Rolle in der Dualitätstheorie von <a href="Operatoralgebra" title="Operatoralgebra">Operatoralgebren</a>. Es bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> die Algebra der <a href="Kompakter_Operator" title="Kompakter Operator">kompakten linearen Operatoren</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">. Jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fcddc7a2f7de54d513a1d4606eba2a9f03ac25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle A\in N(H)}" loading="lazy"> definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{A}(T):=\operatorname {Sp} (AT)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{A}(T):=\operatorname {Sp} (AT)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d9600e8760f2e777e402408d2ddb5647b4f31fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.815ex; height:2.843ex;" alt="{\displaystyle \phi _{A}(T):=\operatorname {Sp} (AT)}" loading="lazy"> ein stetiges lineares Funktional auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy">. Man kann zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)\rightarrow K(H)'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)\rightarrow K(H)'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40f4bdff12c68bed85304c7a68c472603d39a9df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.174ex; height:3.009ex;" alt="{\displaystyle N(H)\rightarrow K(H)'}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mapsto \phi _{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mapsto \phi _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47cded6e7b0943122b6bddfbf08d73eb44446661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.207ex; height:2.509ex;" alt="{\displaystyle A\mapsto \phi _{A}}" loading="lazy"> ein isometrischer Isomorphismus ist, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> mit der nuklearen Norm und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> mit der Operatornorm versehen sei. In diesem Sinne gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)'\cong N(H)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)'\cong N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00e6b551f29f3ecd1960bcf1d2917cb09964f948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.659ex; height:3.009ex;" alt="{\displaystyle K(H)'\cong N(H)}" loading="lazy">.

Genauso definiert jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\in B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7499b1bfd5c275c8e8094319b424a9a7ab9a3aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.114ex; height:2.843ex;" alt="{\displaystyle T\in B(H)}" loading="lazy"> durch die Formel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{T}(A):=\operatorname {Sp} (AT)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{T}(A):=\operatorname {Sp} (AT)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6563f97e8b61fc31c60b8eaaaf250f4a69031314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.974ex; height:2.843ex;" alt="{\displaystyle \psi _{T}(A):=\operatorname {Sp} (AT)}" loading="lazy"> ein stetiges lineares Funktional auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> und man kann wieder zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)\rightarrow N(H)'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)\rightarrow N(H)'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/637a65540fd1bbf68c38d02ecc63666be2e56d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.872ex; height:3.009ex;" alt="{\displaystyle B(H)\rightarrow N(H)'}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mapsto \psi _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\mapsto \psi _{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c612fb34ff49356633049b31bc11ebba5cdc25b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle T\mapsto \psi _{T}}" loading="lazy"> ein isometrischer Isomorphismus ist, wenn man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> mit der nuklearen Norm und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db87fc6eb85eaaaca37af02c2de73bf560157f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle B(H)}" loading="lazy"> mit der Operatornorm versieht. In diesem Sinne gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)'\cong B(H)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)'\cong B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4895a7516a3d8cd4de756428559dd99abfe989cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.357ex; height:3.009ex;" alt="{\displaystyle N(H)'\cong B(H)}" loading="lazy">.

Insbesondere ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)''\cong B(H)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)''\cong B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a44c91aed117771821cf2565a89e7ecb7732eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.811ex; height:3.009ex;" alt="{\displaystyle K(H)''\cong B(H)}" loading="lazy">, das heißt, die Räume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db87fc6eb85eaaaca37af02c2de73bf560157f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle B(H)}" loading="lazy"> sind bei unendlichdimensionalem Hilbertraum nicht <a href="Reflexiver_Raum" title="Reflexiver Raum">reflexiv</a>.

<div class="mw-heading mw-heading2"><h2 id="Eine_Analogie_zu_Folgenräumen">Eine Analogie zu Folgenräumen</h2></div>

Die folgende Aufstellung enthält eine Analogie zwischen <a href="Folgenraum" title="Folgenraum">Folgenräumen</a> komplexer Zahlen und <a href="Operatoralgebra" title="Operatoralgebra">Operatoralgebren</a> auf einem Hilbertraum.

Im Sinne dieser Analogie kann man die nuklearen Operatoren als eine nicht-kommutative Version der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy">-Folgen betrachten, sie ist zumindest eine Merkhilfe.

<th style="width:50%">Folgenraum

</th>

<th>Operatoralgebra

</th></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{00}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e27c5ab99b38e2a5159eaac0f6849f1b6ceb8130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle c_{00}}" loading="lazy"> = Raum der endlichen Folgen</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fa823484cba66c95ba44ec028dc63a87193977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.614ex; height:2.843ex;" alt="{\displaystyle F(H)}" loading="lazy"> = Algebra der <a href="Operator_endlichen_Ranges" title="Operator endlichen Ranges">Operatoren endlichen Ranges</a>

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"> = Raum der Nullfolgen</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> = Algebra der kompakten Operatoren

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> = Raum der absolut summierbaren Folgen</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> = Algebra der nuklearen Operatoren

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy"> = Raum der quadratisch summierbaren Folgen</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle HS(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/834baea30f795bf5aebd28872b31986e4f0cc296.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.436ex; height:2.843ex;" alt="{\displaystyle HS(H)}" loading="lazy"> = Algebra der <a href="Hilbert-Schmidt-Operator" title="Hilbert-Schmidt-Operator">Hilbert-Schmidt-Operatoren</a>

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9c352341ab7260ca1a9004da44c897d1395203c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}" loading="lazy"> = Raum der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-fach summierbaren Folgen, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1<p<\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5540fd86346b4798d7447b8de70fe98cf6243d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty }" loading="lazy"></td>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{p}(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ff5c51ad7dca8d850220ebef2cfa4160b4a4e37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.341ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}_{p}(H)}" loading="lazy"> = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-<a href="Schatten-Klasse" title="Schatten-Klasse">Schatten-Klasse</a>

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"> = Raum der beschränkten Folgen</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db87fc6eb85eaaaca37af02c2de73bf560157f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle B(H)}" loading="lazy"> = Algebra aller beschränkten Operatoren

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{00}\subset \ell ^{1}\subset c_{0}\subset \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48f96238f24b89b9b31d3b3371d7c2eec7327d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.109ex; height:3.009ex;" alt="{\displaystyle c_{00}\subset \ell ^{1}\subset c_{0}\subset \ell ^{\infty }}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(H)\subset N(H)\subset K(H)\subset B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb4e02bdb4b464b7e8fd065476e725d2dd4eba8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.421ex; height:2.843ex;" alt="{\displaystyle F(H)\subset N(H)\subset K(H)\subset B(H)}" loading="lazy">

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{00}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e27c5ab99b38e2a5159eaac0f6849f1b6ceb8130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle c_{00}}" loading="lazy"> liegt dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"> bzgl. der Supremumsnorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b723232adf7317abb1fc1c1326e1e4f79616a7e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.879ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{\infty }}" loading="lazy">.</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fa823484cba66c95ba44ec028dc63a87193977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.614ex; height:2.843ex;" alt="{\displaystyle F(H)}" loading="lazy"> liegt dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> bzgl. der Operatornorm.

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{00}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e27c5ab99b38e2a5159eaac0f6849f1b6ceb8130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle c_{00}}" loading="lazy"> liegt dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> bzgl. der Norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec1a3a7b0d5372a6f2c6d12feda2cea000736bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{1}}" loading="lazy">.</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fa823484cba66c95ba44ec028dc63a87193977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.614ex; height:2.843ex;" alt="{\displaystyle F(H)}" loading="lazy"> liegt dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> bzgl. der nuklearen Norm.

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> liegt dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"> bzgl. der Supremumsnorm.</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> liegt dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> bzgl. der Operatornorm.

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{00}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e27c5ab99b38e2a5159eaac0f6849f1b6ceb8130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle c_{00}}" loading="lazy"> ist ein Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> und in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy">.</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fa823484cba66c95ba44ec028dc63a87193977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.614ex; height:2.843ex;" alt="{\displaystyle F(H)}" loading="lazy"> ist ein Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> und in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db87fc6eb85eaaaca37af02c2de73bf560157f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle B(H)}" loading="lazy">.

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> ist ein Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"> und in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy">.</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy"> ist ein Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> und in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db87fc6eb85eaaaca37af02c2de73bf560157f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle B(H)}" loading="lazy">.

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"> ist ein Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy">.</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431d63f3f8bc9d4124eabb513aa465e63e711eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.939ex; height:2.843ex;" alt="{\displaystyle K(H)}" loading="lazy"> ist ein Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db87fc6eb85eaaaca37af02c2de73bf560157f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle B(H)}" loading="lazy">.

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n}\mapsto \sum _{n=1}^{\infty }a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5bacfa43c0f608aed01daf8eaa481cc060bd85d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.28ex; height:6.843ex;" alt="{\displaystyle (a_{n})_{n}\mapsto \sum _{n=1}^{\infty }a_{n}}" loading="lazy"> ist ein stetiges linearen Funktional auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy">.</td>

<td>Die Spur ist ein stetiges lineares Funktional auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(H)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/240b3889aaa6e730dde8b8a68eba112d8eeb50de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.937ex; height:2.843ex;" alt="{\displaystyle N(H)}" loading="lazy">.

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (c_{0},\|\cdot \|_{\infty })'\cong (\ell ^{1},\|\cdot \|_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa23d71c3925c651590c6b75e8348ae98020f14e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.492ex; height:3.176ex;" alt="{\displaystyle (c_{0},\|\cdot \|_{\infty })'\cong (\ell ^{1},\|\cdot \|_{1})}" loading="lazy">.</td>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K(H),\|\cdot \|)'\cong (N(H),\|\cdot \|_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68f4d15a879190f5f5d2709892ad94ec62774111.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.407ex; height:3.009ex;" alt="{\displaystyle (K(H),\|\cdot \|)'\cong (N(H),\|\cdot \|_{1})}" loading="lazy">.

</td></tr>

<tr>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\ell ^{1},\|\cdot \|_{1})'\cong (\ell ^{\infty },\|\cdot \|_{\infty })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e1cb50c25442ad2d61560841760dbd344e1958d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.276ex; height:3.176ex;" alt="{\displaystyle (\ell ^{1},\|\cdot \|_{1})'\cong (\ell ^{\infty },\|\cdot \|_{\infty })}" loading="lazy">.</td>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (N(H),\|\cdot \|_{1})'\cong (B(H),\|\cdot \|)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80c9e9d438be5dabf39b2705ba2cb0507409251b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.105ex; height:3.009ex;" alt="{\displaystyle (N(H),\|\cdot \|_{1})'\cong (B(H),\|\cdot \|)}" loading="lazy">.

</td></tr>

<tr>

<td>Eine Folge aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"> ist genau dann aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy">, wenn sie das Produkt zweier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy">-Folgen ist.</td>

<td>Ein stetiger linearer Operator ist genau dann nuklear, wenn er das Produkt zweier Hilbert-Schmidt-Operatoren ist.

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Nukleare_Operatoren_auf_Banachräumen">Nukleare Operatoren auf Banachräumen</h2></div>

Die Untersuchung nuklearer Operatoren auf Banachräumen begann 1951 mit einer Arbeit von A. F. Ruston.

Wegen der hier fehlenden Orthonormalbasen sind die Verhältnisse nicht so einfach wie im Hilbertraum-Fall, zudem sind deutlich andere Methoden erforderlich.

Während im Hilbertraum-Fall die Eigenwertfolge eines nuklearen Operators nach obigen Ausführungen absolut summierbar ist, kann man im Banachraum-Fall nur folgende schwächere Aussage beweisen:

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ein Banachraum und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda _{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8a9d843d952fd8553c559b3d9b28d28db711bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.601ex; height:2.843ex;" alt="{\displaystyle (\lambda _{n})_{n}}" loading="lazy"> die Eigenwertfolge eines nuklearen Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in N(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in N(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e2722b0f4dd1e3eade438ed5fd4d1b870f48b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.232ex; height:2.843ex;" alt="{\displaystyle A\in N(E)}" loading="lazy">, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{n})_{n}\in \ell ^{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda _{n})_{n}\in \ell ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea27a53a70ae1f7f69f19f97061f337cdb364f4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.466ex; height:3.176ex;" alt="{\displaystyle (\lambda _{n})_{n}\in \ell ^{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|(\lambda _{n})_{n}\|_{2}\leq \|A\|_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|(\lambda _{n})_{n}\|_{2}\leq \|A\|_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0a3bab3956bc14088f773bc37a6a357abd40dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.201ex; height:2.843ex;" alt="{\displaystyle \|(\lambda _{n})_{n}\|_{2}\leq \|A\|_{1}}" loading="lazy">.

Dieses Ergebnis kann man nicht verbessern; R. J. Kaiser und James Ronald Retherford haben zu vorgegebener <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy">-Folge einen nuklearen Operator aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\ell ^{1}\oplus \ell ^{\infty })}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(\ell ^{1}\oplus \ell ^{\infty })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6aac5c6b8f0a7469308d0e359bc32ee5c29c0c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.582ex; height:3.176ex;" alt="{\displaystyle N(\ell ^{1}\oplus \ell ^{\infty })}" loading="lazy"> mit dieser Eigenwertfolge angegeben. Nach einem Satz von Johnson, König, Maurray und Retherford ist ein Banachraum genau dann isomorph zu einem Hilbertraum, wenn die Eigenwertfolge eines jeden nuklearen Operators aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}" loading="lazy"> ist.

Die Spur eines nuklearen Operators lässt sich nicht für alle Banachräume definieren. Ist eine nukleare Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3b0600a28726bf4d340fc4491687a5f7f785f37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.185ex; height:3.176ex;" alt="{\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}" loading="lazy"> eines Operator aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in N(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in N(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e2722b0f4dd1e3eade438ed5fd4d1b870f48b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.232ex; height:2.843ex;" alt="{\displaystyle A\in N(E)}" loading="lazy"> gegeben, so legt der Hilbertraum-Fall die Definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \operatorname {Sp} (A)=\sum _{n=1}^{\infty }f_{n}(x_{n})}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \operatorname {Sp} (A)=\sum _{n=1}^{\infty }f_{n}(x_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/060e9d1fee0a81cb12ac06e5fb2ad7cf764f1094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.111ex; height:3.176ex;" alt="{\displaystyle \textstyle \operatorname {Sp} (A)=\sum _{n=1}^{\infty }f_{n}(x_{n})}" loading="lazy"> nahe. Diese Zahl erweist sich genau dann als wohldefiniert, das heißt als unabhängig von der gewählten nuklearen Darstellung, wenn der Banachraum die <a href="Approximationseigenschaft" title="Approximationseigenschaft">Approximationseigenschaft</a> hat.

Die im Hilbertraum-Fall vorliegende Dualität verallgemeinert sich wie folgt auf Banachräume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> mit Approximationseigenschaft. Jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in B(E,E'')}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\in B(E,E'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3740d3853b7234534d20f277cad19712420ca8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.791ex; height:3.009ex;" alt="{\displaystyle T\in B(E,E'')}" loading="lazy"> definiert ein stetiges lineares Funktional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bcedcb4331027779fdef6214e4e42f072b0f903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.902ex; height:2.509ex;" alt="{\displaystyle \psi _{T}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbe797df2434f2e14ced1b8d3440dbd580799b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.649ex; height:2.843ex;" alt="{\displaystyle N(E)}" loading="lazy"> mittels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \psi _{T}(A)=\sum _{n=1}^{\infty }(T(x_{n}))(f_{n})}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \psi _{T}(A)=\sum _{n=1}^{\infty }(T(x_{n}))(f_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/972a8d84349eadee744c3eb6450fbb9a323dfa3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.296ex; height:3.176ex;" alt="{\displaystyle \textstyle \psi _{T}(A)=\sum _{n=1}^{\infty }(T(x_{n}))(f_{n})}" loading="lazy">, worin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3b0600a28726bf4d340fc4491687a5f7f785f37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.185ex; height:3.176ex;" alt="{\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}" loading="lazy"> eine nukleare Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ist. Die Approximationseigenschaft sichert die Wohldefiniertheit, d. h. die Unabhängigkeit von der Wahl der nuklearen Darstellung. Man kann zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(E,E'')\rightarrow N(E)'}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(E,E'')\rightarrow N(E)'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d2a72abf2f1d7402a19444e9f0063343d0e7202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.261ex; height:3.009ex;" alt="{\displaystyle B(E,E'')\rightarrow N(E)'}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mapsto \psi _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\mapsto \psi _{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c612fb34ff49356633049b31bc11ebba5cdc25b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle T\mapsto \psi _{T}}" loading="lazy"> ein isometrischer Isomorphismus ist, wenn man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbe797df2434f2e14ced1b8d3440dbd580799b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.649ex; height:2.843ex;" alt="{\displaystyle N(E)}" loading="lazy"> mit der nuklearen Norm und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(E,E'')}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(E,E'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99bbba3d1ea6916262507a2dbd98d322fb7488bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.314ex; height:3.009ex;" alt="{\displaystyle B(E,E'')}" loading="lazy"> mit der Operatornorm versieht. In diesem Sinne ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E)'\cong B(E,E'')}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E)'\cong B(E,E'')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0786080e9e3f37850786b8d7b6d8eebdc742c456.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.746ex; height:3.009ex;" alt="{\displaystyle N(E)'\cong B(E,E'')}" loading="lazy">. Ist daher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> zusätzlich <a href="Reflexiver_Raum" title="Reflexiver Raum">reflexiv</a>, so hat man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(E)'\cong B(E)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(E)'\cong B(E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e31c0e45ca02070f3dc55f0f0d61747df061de9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.781ex; height:3.009ex;" alt="{\displaystyle N(E)'\cong B(E)}" loading="lazy"> wie im Hilbertraum-Fall.

<div class="mw-heading mw-heading2"><h2 id="Nukleare_Operatoren_auf_lokalkonvexen_Räumen">Nukleare Operatoren auf lokalkonvexen Räumen</h2></div>

Alexander Grothendieck hat 1951 mit der Untersuchung nuklearer Operatoren zwischen lokalkonvexen Räumen begonnen.

Da man auf lokalkonvexen Räumen keine Norm zur Verfügung hat, muss die Definition wie folgt formuliert werden: Ein linearer Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:E\rightarrow F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:E\rightarrow F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93ff6b7547b10ceab4a51f1c5e021a8d7c6a56a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.811ex; height:2.176ex;" alt="{\displaystyle A:E\rightarrow F}" loading="lazy"> heißt nuklear, falls es eine Darstellung der Art

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x)=\sum _{n=1}^{\infty }\lambda _{n}f_{n}(x)y_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cf11ad0b7fb0d39d159c8654d41eeeb27dd29d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.151ex; height:6.843ex;" alt="{\displaystyle A(x)=\sum _{n=1}^{\infty }\lambda _{n}f_{n}(x)y_{n}}" loading="lazy">

gibt, wobei

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda _{n})_{n}\in \ell ^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c15bbde464842696088b79788123108f150e685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.466ex; height:3.176ex;" alt="{\displaystyle (\lambda _{n})_{n}\in \ell ^{1}}" loading="lazy">,</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a447cb7ffab7af3e68d8c04ee2a1c38059ef135f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle (f_{n})_{n}}" loading="lazy"> eine gleichstetige Folge im <a href="Dualraum" title="Dualraum">starken Dualraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a574600572696493d48300245a45b8de0638ce21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}" loading="lazy"> ist (d. h., es gibt eine stetige <a href="Halbnorm" title="Halbnorm">Halbnorm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f_{n}(x)|\leq p(x)}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f_{n}(x)|\leq p(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac5e33bd405c6bd9dcb0f6afb3e6c5d67ab0205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.197ex; height:2.843ex;" alt="{\displaystyle |f_{n}(x)|\leq p(x)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30b1971b01bc31d5b816f03cc7e1d9215d6c2ad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.946ex; height:2.176ex;" alt="{\displaystyle x\in E}" loading="lazy">),</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y_{n})_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b66bae98983d019aa2b9995e7049c28b0da2ed77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle (y_{n})_{n}}" loading="lazy"> eine <a href="Beschr%C3%A4nktheit" class="mw-redirect" title="Beschränktheit">beschränkte</a> Folge in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> ist.</li></ul>

Da die geforderte Gleichstetigkeit im Banachraum-Fall der Beschränktheit gleichkommt, führt die hier gegebene Definition im Banachraum-Fall auf denselben Begriff des nuklearen Operators, wie er oben definiert wurde.

Die Ideal-Eigenschaft verallgemeinert sich auf lokalkonvexe Räume: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon E\rightarrow F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon E\rightarrow F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dbf1509c61519690d814ab770462cea64e61bca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.908ex; height:2.176ex;" alt="{\displaystyle A\colon E\rightarrow F}" loading="lazy"> nuklear und sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\colon G\rightarrow E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\colon G\rightarrow E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1227f8155968bd2a03e1a4c0f5c490325fb93c82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.014ex; height:2.176ex;" alt="{\displaystyle B\colon G\rightarrow E}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\colon F\rightarrow H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\colon F\rightarrow H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2d093296f7d2428cd70a6e6f5cabd73fee9a6cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.219ex; height:2.176ex;" alt="{\displaystyle C\colon F\rightarrow H}" loading="lazy"> stetige, lineare Operatoren zwischen lokalkonvexen Räumen, so ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CAB\colon G\rightarrow H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle CAB\colon G\rightarrow H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e307ed36e81011e2a083fcba99b2a225c67bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.812ex; height:2.176ex;" alt="{\displaystyle CAB\colon G\rightarrow H}" loading="lazy"> nuklear. Nukleare Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon E\rightarrow F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon E\rightarrow F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dbf1509c61519690d814ab770462cea64e61bca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.908ex; height:2.176ex;" alt="{\displaystyle A\colon E\rightarrow F}" loading="lazy"> sind stetig und, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> <a href="Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum">vollständig</a> ist, sogar kompakt.

Man kann zeigen, dass es zu jedem nuklearen Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon E\rightarrow F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon E\rightarrow F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dbf1509c61519690d814ab770462cea64e61bca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.908ex; height:2.176ex;" alt="{\displaystyle A\colon E\rightarrow F}" loading="lazy"> einen weiteren nuklearen Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {A}}\colon G\rightarrow H}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}\colon G\rightarrow H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55fb03aa7d1a54316189101d674c55cf51caf897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.314ex; height:2.676ex;" alt="{\displaystyle {\tilde {A}}\colon G\rightarrow H}" loading="lazy"> zwischen normierten Räumen und stetige lineare Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\colon E\rightarrow G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\colon E\rightarrow G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/127e7ebb5daa94e90309bdf41ef833b05c435630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.014ex; height:2.176ex;" alt="{\displaystyle B\colon E\rightarrow G}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\colon H\rightarrow F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\colon H\rightarrow F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b9746e48593ccc7cd4e968ed908cd5821d2b26b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.219ex; height:2.176ex;" alt="{\displaystyle C\colon H\rightarrow F}" loading="lazy"> gibt mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=C{\tilde {A}}B}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=C{\tilde {A}}B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/566f902b9f7128a90f9ee730d2a5e3af3bb23803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.148ex; height:2.676ex;" alt="{\displaystyle A=C{\tilde {A}}B}" loading="lazy">. Damit kann man das Studium der nuklearen Operatoren zwischen lokalkonvexen Räumen auf den normierten Fall zurückführen.

In der lokalkonvexen Theorie spielen die nuklearen Operatoren eine wichtige Rolle im Zusammenhang mit <a href="Nuklearer_Raum" title="Nuklearer Raum">nuklearen Räumen</a>.

<div class="mw-heading mw-heading2"><h2 id="Anwendung_in_der_statistischen_Physik">Anwendung in der statistischen Physik</h2></div>

Das physikalische Gebiet der <a href="Statistische_Physik" title="Statistische Physik">Statistischen Physik</a> beruht auf der zentralen Annahme, dass die Spur jeder mit der Exponentialfunktion des sog. <a href="Hamilton-Operator" class="mw-redirect" title="Hamilton-Operator">Hamilton-Operators</a> (Energieoperator) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"> bei der Temperatur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> gewichteten Messgröße (<a href="Observable" title="Observable">Observable</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f595a6c73d1183d6a1b2ac21fe47ac28c1483821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}" loading="lazy"> der <a href="Quantenstatistik" title="Quantenstatistik">Quantenstatistik</a> existiert, und zwar obwohl der Hamiltonoperator selbst keineswegs zur Spurklasse gehört und in der Regel auch für den (nur selbstadjungierten!) Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f595a6c73d1183d6a1b2ac21fe47ac28c1483821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}" loading="lazy"> dasselbe zutrifft. Für den thermischen Erwartungswert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\hat {A}}\rangle _{T}}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle {\hat {A}}\rangle _{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8379930ca9252e7bb5813d1ba2b28ae77dcd6e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.974ex; height:3.343ex;" alt="{\displaystyle \langle {\hat {A}}\rangle _{T}}" loading="lazy"> der betrachteten Messgröße gilt trotzdem aufgrund dieser Annahme die Beziehung

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msup>

<mfrac>

</mrow>

</mfrac>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle {\hat {A}}\rangle _{T}=\operatorname {Sp} \{e^{-{\frac {\mathcal {H}}{T}}}\,{\hat {A}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4705f61e6d381e369227e0984cdd2b35b0b445e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.703ex; height:4.176ex;" alt="{\displaystyle \langle {\hat {A}}\rangle _{T}=\operatorname {Sp} \{e^{-{\frac {\mathcal {H}}{T}}}\,{\hat {A}}\}}" loading="lazy">. </dd></dl>

Anders gesagt: Die eingeklammerten Ausdrücke befassen sich i. W. mit nuklearen Räumen und den darin definierten Operatoren bzw. Messgrößen.

<div class="mw-heading mw-heading2"><h2 id="Schur-Basis_in_einem_Hilbert-Raum">Schur-Basis in einem Hilbert-Raum</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{p}:=\{A:\operatorname {Sp} (|A|^{p})<\infty \}}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{p}:=\{A:\operatorname {Sp} (|A|^{p})<\infty \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09d6d505e90f79e84b955e7462515ae8947effe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.131ex; height:3.176ex;" alt="{\displaystyle {\mathcal {S}}_{p}:=\{A:\operatorname {Sp} (|A|^{p})<\infty \}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/147ac20997a4d9a8eeb88fbc23bc47107268a97e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle N(A)}" loading="lazy"> die Anzahl der nichttrivialen Eigenwerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(A):=\vert \{n:\lambda _{n}(A)\neq 0\}\vert }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(A):=\vert \{n:\lambda _{n}(A)\neq 0\}\vert }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53d07dd1f80962a1e1092acd4ae4d3ea9e6f7171.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.699ex; height:2.843ex;" alt="{\displaystyle N(A):=\vert \{n:\lambda _{n}(A)\neq 0\}\vert }" loading="lazy">. Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {S}}_{\infty }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {S}}_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ccaac4ecc5182a22d27892a4bd591af69fe0dd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.868ex; height:2.509ex;" alt="{\displaystyle A\in {\mathcal {S}}_{\infty }}" loading="lazy">, dann nennen manche Autoren (z. B. <a href="Barry_Simon" title="Barry Simon">Barry Simon</a>) die orthonormale Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\eta _{n}\}_{n=1}^{N(A)}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\eta _{n}\}_{n=1}^{N(A)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bafdcac5d7b14b73c41fd357b8d70cd43d0c3c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.902ex; height:3.676ex;" alt="{\displaystyle \{\eta _{n}\}_{n=1}^{N(A)}}" loading="lazy"> mit

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{n}(A)=\langle \eta _{n},A\eta _{n}\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dd0eb90e6bf17c64e2258422ec80afa99e96a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.559ex; height:2.843ex;" alt="{\displaystyle \lambda _{n}(A)=\langle \eta _{n},A\eta _{n}\rangle }" loading="lazy"></dd></dl>

Schur-Basis.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Klaus_Floret" title="Klaus Floret">Klaus Floret</a>, <a href="Joseph_Wloka" title="Joseph Wloka">Joseph Wloka</a>: Einführung in die Theorie der lokalkonvexen Räume (= <a href="Lecture_Notes_in_Mathematics" title="Lecture Notes in Mathematics">Lecture Notes in Mathematics</a>. 56). Springer, Berlin u. a. 1968.</li>

<li><a href="Alexandre_Grothendieck" class="mw-redirect" title="Alexandre Grothendieck">Alexandre Grothendieck</a>: Sur une notion de produit tensoriel topologique d’espaces vectoriels topologiques, et une classe remarquable d’espaces vectoriels liée à cette notion. In: Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Paris. Band 233, 1951, <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k3185k/f1556.item">S. 1556–1558</a>.</li>

<li>R. J. Kaiser, Joel R. Retherford: Preassigning eigenvalues and zeros of nuclear operators. In: <a href="Studia_Mathematica" title="Studia Mathematica">Studia Mathematica</a>. Band 81, 1985, S. 127–133, (<a rel="nofollow" class="external text" href="https://bibliotekanauki.pl/articles/1388807">online</a>).</li>

<li><a href="Albrecht_Pietsch" title="Albrecht Pietsch">Albrecht Pietsch</a>: Eigenvalues and s-Numbers (= Cambridge Studies in Advanced Mathematics. 13). Cambridge University Press, Cambridge u. a. 1987, ISBN 0-521-32532-3.</li>

<li>Albrecht Pietsch: Nukleare lokalkonvexe Räume (= Schriftenreihe der Institute für Mathematik bei der Deutschen Akademie der Wissenschaften zu Berlin. Reihe A: Reine Mathematik. 1, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220568-4188%22&key=cql">0568-4188</a>). Akademie-Verlag, Berlin 1965.</li>

<li>Reinhold Meise, Dietmar Vogt: Einführung in die Funktionalanalysis (= Vieweg-Studium. 62, Aufbaukurs Mathematik.). Vieweg, Braunschweig u. a. 1992, ISBN 3-528-07262-8.</li>

<li>Anthony F. Ruston: On the Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space. In: <a href="Proceedings_of_the_London_Mathematical_Society" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a>. Serie 2, Band 53, Nummer 2, 1951, S. 109–124, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112/plms%2Fs2-53.2.109">10.1112/plms/s2-53.2.109</a>.</li>

<li><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Helmut H. Schaefer</a>: Topological Vector Spaces (= <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. 3). 3rd printing corrected. Springer, New York NY u. a. 1971, ISBN 0-387-90026-8.</li>

<li><a href="Robert_Schatten" title="Robert Schatten">Robert Schatten</a>, <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a>: The Cross Space of Linear Transformations. II. In: <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. Serie 2, Band 47, Nummer 3, 1946, S. 608–630, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/1969096">1969096</a>.</li>

<li><a href="Dirk_Werner_(Mathematiker)" title="Dirk Werner (Mathematiker)">Dirk Werner</a>: Funktionalanalysis. 2., überarbeitete Auflage. Springer, Berlin u. a. 1997, ISBN 3-540-61904-6.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Barry_Simon" title="Barry Simon">Barry Simon</a>: Notes on Infinite Determinants of Hilbert Space Operators. In: <a href="Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a>. Band 24, Nummer 3, 1977, S. 244–273, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016/0001-8708%2877%2990057-3">10.1016/0001-8708(77)90057-3</a>.

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-11" href="https://de.wikipedia.org/wiki/?title=Spurklasseoperator&oldid=262324194">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>