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<title>Fredholm-Operator</title>

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

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<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">In der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>, einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>, ist die Klasse der Fredholm-Operatoren (nach <a href="Erik_Ivar_Fredholm" title="Erik Ivar Fredholm">E. I. Fredholm</a>) eine bestimmte Klasse <a href="Linearer_Operator" title="Linearer Operator">linearer Operatoren</a>, die man „fast“ invertieren kann. Jedem Fredholm-Operator ordnet man eine <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahl</a> zu, diese wird Fredholm-Index, analytischer Index oder kurz Index genannt. Der Fredholm-Operator wird auch Noether-Operator nach dem Mathematiker <a href="Fritz_Noether" title="Fritz Noether">Fritz Noether</a> genannt. Dieser hatte als erster Operatoren mit endlichem, aber nicht verschwindendem Index untersucht.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

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

Ein beschränkter <a href="Linearer_Operator" title="Linearer Operator">linearer Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon X\to Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3fcb04477030783ab411037da34caeec858b0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.144ex; height:2.176ex;" alt="{\displaystyle A\colon X\to Y}" loading="lazy"> zwischen zwei <a href="Banachraum" title="Banachraum">Banachräumen</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> heißt Fredholm-Operator, oder man sagt kurz: "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</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 Fredholm", wenn

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2389174655546bb240cbde9862cc5253fc67de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.302ex; height:2.176ex;" alt="{\displaystyle \ker A}" loading="lazy"> endliche <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> hat und</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d76c23030184ac0828b8605fc28b71711c0dde4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.497ex; height:2.176ex;" alt="{\displaystyle \mathrm {ran} \,A}" loading="lazy"> endliche <a href="Kodimension" title="Kodimension">Kodimension</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> hat.</li></ul>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2389174655546bb240cbde9862cc5253fc67de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.302ex; height:2.176ex;" alt="{\displaystyle \ker A}" loading="lazy"> der <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</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 die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in X:Ax=0\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x\in X:Ax=0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7e3bf716f57e853187ed5c5398537bc9af0ed58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.746ex; height:2.843ex;" alt="{\displaystyle \{x\in X:Ax=0\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ran} \,A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d76c23030184ac0828b8605fc28b71711c0dde4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.497ex; height:2.176ex;" alt="{\displaystyle \mathrm {ran} \,A}" loading="lazy"> ist das <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Bild</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</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 die Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Ax\mid x\in X\}\subseteq Y}">

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<annotation encoding="application/x-tex">{\displaystyle \{Ax\mid x\in X\}\subseteq Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/407d9e9718f2a8f251cc0620d5f5b2968d442b03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.357ex; height:2.843ex;" alt="{\displaystyle \{Ax\mid x\in X\}\subseteq Y}" loading="lazy">.

Die Zahl

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ind} (A)=\dim(\ker A)-\mathrm {codim} (\mathrm {ran} \,A,Y)\in \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d799a90916f2b3ff7da32874413a86ed93e4bd90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.284ex; height:2.843ex;" alt="{\displaystyle \mathrm {ind} (A)=\dim(\ker A)-\mathrm {codim} (\mathrm {ran} \,A,Y)\in \mathbb {Z} }" loading="lazy"></dd></dl>

heißt Fredholm-Index von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</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">.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_des_Fredholm-Indexes">Eigenschaften des Fredholm-Indexes</h2></div>

Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (X,Y)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9b31260068a13946176a3cba5b1ebb42a7dad78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.275ex; height:2.843ex;" alt="{\displaystyle \Phi (X,Y)}" loading="lazy"> der Fredholm-Operatoren zwischen den Banachräumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> ist <a href="Offene_Menge" title="Offene Menge">offen</a> in der Menge der beschränkten Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(X,Y)}">

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<annotation encoding="application/x-tex">{\displaystyle L(X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d06d419d8c386f560a7de41b39972a6c55c55141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.179ex; height:2.843ex;" alt="{\displaystyle L(X,Y)}" loading="lazy">. Auf jeder Zusammenhangskomponente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (X,Y)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9b31260068a13946176a3cba5b1ebb42a7dad78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.275ex; height:2.843ex;" alt="{\displaystyle \Phi (X,Y)}" loading="lazy"> ist der Fredholm-Index konstant, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ind} (A)=n_{U}\in \mathbb {Z} }">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ind} (A)=n_{U}\in \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4afd63bcb7d851ff64370c19c57c32dfb159568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.161ex; height:2.843ex;" alt="{\displaystyle \operatorname {ind} (A)=n_{U}\in \mathbb {Z} }" loading="lazy"> für alle Fredholm-Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/814abd0f646c3349e4e794935c44321e5d2f2eed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.366ex; height:2.176ex;" alt="{\displaystyle A\in U}" loading="lazy">. Tatsächlich ist die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto n_{U}}">

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<annotation encoding="application/x-tex">{\displaystyle U\mapsto n_{U}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17ffe3df34ffffaecdbf76e2b3a0d7e5cff544fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.284ex; height:2.509ex;" alt="{\displaystyle U\mapsto n_{U}}" loading="lazy"> von der Zusammenhangskomponente auf den Index bijektiv. Daraus ergeben sich die folgenden Eigenschaften des Index:

<ul><li>Die Indexabbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ind} \colon \Phi (X,Y)\to \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ind} \colon \Phi (X,Y)\to \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16e683cb287486be0d65a111cbc279f3ae7c309a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.705ex; height:2.843ex;" alt="{\displaystyle \operatorname {ind} \colon \Phi (X,Y)\to \mathbb {Z} }" loading="lazy"> ist stetig.</li>

<li>Der Index ist invariant unter kleinen Störungen, das heißt, zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Phi (X,Y)}">

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<annotation encoding="application/x-tex">{\displaystyle A\in \Phi (X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d4380626093d719ee94674b1e64016f829ab1aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.858ex; height:2.843ex;" alt="{\displaystyle A\in \Phi (X,Y)}" loading="lazy"> gibt es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy">, so dass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in L(X,Y)}">

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<annotation encoding="application/x-tex">{\displaystyle S\in L(X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ff8cbf4342b7a0194eafbbb8f8e51206a7239c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.519ex; height:2.843ex;" alt="{\displaystyle S\in L(X,Y)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|S\|<\varepsilon }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f6ac1ab1902adda6031534f9556673362a9a895.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \|S\|<\varepsilon }" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ind} (A+S)=\operatorname {ind} (A)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edaf85eac3b66ee4393349794502fd04d2c38853.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.007ex; height:2.843ex;" alt="{\displaystyle \operatorname {ind} (A+S)=\operatorname {ind} (A)}" loading="lazy">.</li>

<li>Der Index ist eine <a href="Homotopieinvariante" class="mw-redirect" title="Homotopieinvariante">homotopie-invariante</a> Zahl.<a href="#cite_note-4">[4]</a>: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \colon [0,1]\to \Phi (X,Y)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi \colon [0,1]\to \Phi (X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2641a8ce60b84f72dddd19b50ca56e6a362b352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.088ex; height:2.843ex;" alt="{\displaystyle \psi \colon [0,1]\to \Phi (X,Y)}" loading="lazy"> stetig, dann haben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcfdf3fefa6feee8b2192815ac15a2650d8a8db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle \psi (0)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (1)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df731c3f72ce04120fd347ee347472e8d854bbbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle \psi (1)}" loading="lazy"> den gleichen Index.</li>

<li>Der Fredholm-Index, als Abbildung von der Menge der Fredholm-Operatoren in die Menge der ganzen Zahlen, ist <a href="Surjektive_Abbildung" class="mw-redirect" title="Surjektive Abbildung">surjektiv</a>.<a href="#cite_note-5">[5]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_des_Fredholm-Operatores">Eigenschaften des Fredholm-Operatores</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Bild_ist_abgeschlossener_Unterraum">Bild ist abgeschlossener Unterraum</h3></div>

Das Bild <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ran} (A)}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b91cdb957d633eb8cb708c60b2b6ece3eb9b3e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.919ex; height:2.843ex;" alt="{\displaystyle \mathrm {ran} (A)}" loading="lazy"> eines Fredholm-Operators ist ein abgeschlossener Unterraum.

<div class="mw-heading mw-heading3"><h3 id="Struktur">Struktur</h3></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3fcb04477030783ab411037da34caeec858b0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.144ex; height:2.176ex;" alt="{\displaystyle A\colon X\to Y}" loading="lazy"> ein Fredholm-Operator, dann hat der endlich-dimensionale Unterraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker(A)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b8ae78a354993696bfca512586a9d0c2e8141b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.724ex; height:2.843ex;" alt="{\displaystyle \ker(A)}" loading="lazy"> einen abgeschlossenen Komplementärraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, d. h., es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\ker(A)\oplus W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\ker(A)\oplus W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92b27c03ab0adb11eff5af0ad2f3d7f7b06cfcd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.078ex; height:2.843ex;" alt="{\displaystyle X=\ker(A)\oplus W}" loading="lazy">. Die Einschränkung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {A}}\colon W\to \operatorname {ran} (A)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {A}}\colon W\to \operatorname {ran} (A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c109653bd4e2df0fa988b9c5b938867378000fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.843ex; height:3.343ex;" alt="{\displaystyle {\widetilde {A}}\colon W\to \operatorname {ran} (A)}" loading="lazy"> 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 dann offenbar ein bijektiver Operator, dessen Inverse nach dem <a href="Satz_%C3%BCber_die_offene_Abbildung#Satz_vom_stetigen_Inversen" class="mw-redirect" title="Satz über die offene Abbildung">Satz vom stetigen Inversen</a> ebenfalls beschränkt ist. Der Operator <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 also "bis auf endlich viele Dimensionen" stetig invertierbar. Viele der folgenden Eigenschaften lassen sich damit beweisen.

<div class="mw-heading mw-heading3"><h3 id="Komposition">Komposition</h3></div>

Die Komposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\circ B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c341d3106d2763836b32f992b74e73f4cef0d24d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.702ex; height:2.176ex;" alt="{\displaystyle A\circ B}" loading="lazy"> zweier Fredholm-Operatoren <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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ist wieder ein Fredholm-Operator und für den Index gilt<a href="#cite_note-6">[6]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ind} (A\circ B)=\operatorname {ind} (A)+\operatorname {ind} (B)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/211b5c2e3fac834cf8398c9ed1a40b4d596bea4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.271ex; height:2.843ex;" alt="{\displaystyle \operatorname {ind} (A\circ B)=\operatorname {ind} (A)+\operatorname {ind} (B)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Dualer_Operator">Dualer Operator</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'\colon Y'\to X'}">

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'\colon Y'\to X'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6fb5a529c53d1a7119e29996b2b0771a8d7e7fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.342ex; height:2.509ex;" alt="{\displaystyle A'\colon Y'\to X'}" loading="lazy"> der zum Fredholm-Operator <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"> <a href="Dualer_Operator" class="mw-redirect" title="Dualer Operator">duale Operator</a>. Dann gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\ker A')=\operatorname {codim} (\mathrm {ran} (A))}">

<msup>

</msup>

<mi>codim</mi>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim(\ker A')=\operatorname {codim} (\mathrm {ran} (A))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d89f756d019b4a51e42136d0626e679fcb2821d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.568ex; height:3.009ex;" alt="{\displaystyle \dim(\ker A')=\operatorname {codim} (\mathrm {ran} (A))}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {codim} (\mathrm {ran} (A'))=\dim(\ker A)}">

<mi>codim</mi>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {codim} (\mathrm {ran} (A'))=\dim(\ker A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f4c39e18238c1a61ee938099ea2d8178c4b67df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.568ex; height:3.009ex;" alt="{\displaystyle \operatorname {codim} (\mathrm {ran} (A'))=\dim(\ker A)}" loading="lazy">. Daher ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"> ein Fredholm-Operator und für seinen Index gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ind} (T')=-\operatorname {ind} (T)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ind} (T')=-\operatorname {ind} (T)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00466c168f0bb40748fd9243c9c540b10bfb01bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.417ex; height:3.009ex;" alt="{\displaystyle \operatorname {ind} (T')=-\operatorname {ind} (T)}" loading="lazy">.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Satz_von_Atkinson">Satz von Atkinson</h3></div>

Nach dem <a href="Satz_von_Atkinson" class="mw-redirect" title="Satz von Atkinson">Satz von Atkinson</a> ist ein Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3fcb04477030783ab411037da34caeec858b0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.144ex; height:2.176ex;" alt="{\displaystyle A\colon X\to Y}" loading="lazy"> genau dann ein Fredholm-Operator, wenn es Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1},B_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{1},B_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c773f341a6a6e72f195fbb73c87dc781482a521c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.67ex; height:2.509ex;" alt="{\displaystyle B_{1},B_{2}}" loading="lazy"> und <a href="Kompakter_Operator" title="Kompakter Operator">kompakte Operatoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1},K_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{1},K_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20e1ca4f18aed738f059171d774a63a07835d325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.088ex; height:2.509ex;" alt="{\displaystyle K_{1},K_{2}}" loading="lazy"> gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB_{1}=I_{Y}-K_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AB_{1}=I_{Y}-K_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2587bf7247760fceacaa50f5e3915a0a6292a70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.037ex; height:2.509ex;" alt="{\displaystyle AB_{1}=I_{Y}-K_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2}A=I_{X}-K_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{2}A=I_{X}-K_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30dfb66e6654d5fab7312eb9d6230899376417d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.183ex; height:2.509ex;" alt="{\displaystyle B_{2}A=I_{X}-K_{2}}" loading="lazy"> gilt, das heißt wenn <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"> modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon X\to X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba5a75e57f748d59da24ed0522c122db3eb9786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.351ex; height:2.176ex;" alt="{\displaystyle A\colon X\to X}" loading="lazy"> genau dann ein Fredholm-Operator, wenn seine Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A]_{{\mathcal {C}}(X)}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [A]_{{\mathcal {C}}(X)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b371e954f0d41ce8139d9a18394c619a69c6853c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.825ex; height:3.176ex;" alt="{\displaystyle [A]_{{\mathcal {C}}(X)}}" loading="lazy"> in der <a href="Calkin-Algebra" title="Calkin-Algebra">Calkin-Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(X)/{\mathcal {C}}(X)}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a06e8f5e1bfc7274207b785344593f81cef3d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.524ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(X)/{\mathcal {C}}(X)}" loading="lazy"> invertierbar ist.

<div class="mw-heading mw-heading3"><h3 id="Kompakte_Störung">Kompakte Störung</h3></div>

Für jeden Fredholm-Operator <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"> und jeden kompakten Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79f60dda90f6c7d4586b0be0ba29936a6f536573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.649ex; height:2.343ex;" alt="{\displaystyle A+K}" loading="lazy"> ebenfalls ein Fredholm-Operator mit gleichem Fredholm-Index wie <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">. Daher sagt man, dass der Index eines Fredholm-Operators invariant unter kompakten Störungen ist. Insbesondere ist jede kompakte Störung der Identität, also jeder Operator der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I+K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I+K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffbb49de172be15aef5139aacbbdf9587431996b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.078ex; height:2.343ex;" alt="{\displaystyle I+K}" loading="lazy"> für einen kompakten Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein Fredholm-Operator vom Index 0.

<div class="mw-heading mw-heading3"><h3 id="Punctured_Neighbourhood_Theorem">Punctured Neighbourhood Theorem</h3></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba5a75e57f748d59da24ed0522c122db3eb9786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.351ex; height:2.176ex;" alt="{\displaystyle A\colon X\to X}" loading="lazy"> ein Fredholm-Operator, dann gibt es nach dem Punctured Neighbourhood Theorem ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy">, so dass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11a6d1585381827bdf73529c2a418bc14098567c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {C} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<|\lambda |<\varepsilon }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fa628c6108dce5cab45fb0a7f10b904d3c18e0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.092ex; height:2.843ex;" alt="{\displaystyle 0<|\lambda |<\varepsilon }" loading="lazy">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim \ker(A-\lambda I)\equiv \mathrm {const} \leq \dim \ker A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2581d43501c2e849fbfa4565addff60958d35acf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.423ex; height:2.843ex;" alt="{\displaystyle \dim \ker(A-\lambda I)\equiv \mathrm {const} \leq \dim \ker A}" loading="lazy"> und</li>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {codim} \,\mathrm {ran} (A-\lambda I)\equiv \mathrm {const} \leq \mathrm {codim} \,\mathrm {ran} A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5769755f44738d438e23fbf89392c013bfd13c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.816ex; height:2.843ex;" alt="{\displaystyle \mathrm {codim} \,\mathrm {ran} (A-\lambda I)\equiv \mathrm {const} \leq \mathrm {codim} \,\mathrm {ran} A}" loading="lazy"></li></ol>

gilt.<a href="#cite_note-8">[8]</a> Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ba67e098a60d3285b10f2e76be219fc3c52d690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.11ex; height:2.343ex;" alt="{\displaystyle A-\lambda I}" loading="lazy"> also ein Fredholm-Operator. Da der Fredholm-Index stetig ist, folgt daraus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ind} (A-\lambda I)=\mathrm {ind} (A)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ind} (A-\lambda I)=\mathrm {ind} (A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd3bcad25f24198d330d46ad96ced63211af915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.034ex; height:2.843ex;" alt="{\displaystyle \mathrm {ind} (A-\lambda I)=\mathrm {ind} (A)}" loading="lazy">. Bewiesen wurde das Punctured Neighbourhood Theorem von <a href="Israel_Gohberg" title="Israel Gohberg">Israel Gohberg</a>.<a href="#cite_note-9">[9]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Shiftoperator">Shiftoperator</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Shiftoperator" title="Shiftoperator">Shiftoperator</a></div>

<div class="mw-heading mw-heading3"><h3 id="Integraloperator">Integraloperator</h3></div>

Ein klassisches Beispiel eines Fredholm-Operators ist der Operator

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcb81eeea762c30a9d5540b57ca547288f46e125.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.137ex; height:2.343ex;" alt="{\displaystyle A:=I+T}" loading="lazy">,</dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> der Identitätsoperator und <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"> ein <a href="Kompakter_Operator" title="Kompakter Operator">kompakter Operator</a> ist. Auf dem Banachraum der stetigen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C([a,b])}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C([a,b])}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8e4f44fa2823fcdffc5fc26981c0d4fa57cade9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.131ex; height:2.843ex;" alt="{\displaystyle C([a,b])}" loading="lazy"> beziehungsweise auf dem der <a href="Quadratintegrierbare_Funktion" class="mw-redirect" title="Quadratintegrierbare Funktion">quadratintegrierbaren Funktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}([a,b])}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}([a,b])}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9afbae301dd78d6739e6c45a942076bf669595e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.001ex; height:3.176ex;" alt="{\displaystyle L^{2}([a,b])}" loading="lazy"> ist der Operator <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"> von der Form

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\phi )(x)=\phi (x)+\int _{a}^{b}k(x,y)\phi (y)\mathrm {d} y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2961a7e2effa181c97306e7c396307915b2b064.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.667ex; height:6.343ex;" alt="{\displaystyle (A\phi )(x)=\phi (x)+\int _{a}^{b}k(x,y)\phi (y)\mathrm {d} y}" loading="lazy">,</dd></dl>

wobei der <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> eine stetige beziehungsweise quadratintegrierbare Funktion ist. Dieser Fredholm-Operator hat den Index 0. In der Fredholm-Theorie werden Gleichungen des Typs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\phi (x)=f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bad3327fb905a8413b3589bd170ed2003558f070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.784ex; height:2.843ex;" alt="{\displaystyle A\phi (x)=f(x)}" loading="lazy"> untersucht. Die <a href="Fredholm-Alternative" class="mw-redirect" title="Fredholm-Alternative">Fredholm-Alternative</a> als ein zentrales Resultat der Fredholm-Theorie gibt eine Antwort, unter welchen Bedingungen Gleichungen dieses Typs lösbar sind.

<div class="mw-heading mw-heading3"><h3 id="Laplace-Operator">Laplace-Operator</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a></div>

Der Laplace-Operator

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=\sum _{k=1}^{n}{\partial ^{2}f \over \partial x_{k}^{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/002372419c135d8c74a266e77a0fd1efb91e0e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.628ex; height:6.843ex;" alt="{\displaystyle \Delta f=\sum _{k=1}^{n}{\partial ^{2}f \over \partial x_{k}^{2}}}" loading="lazy"></dd></dl>

definiert auf dem <a href="Sobolev-Raum" title="Sobolev-Raum">Sobolev-Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{2}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d32a7f4dbd47d09b1196c345d844bbfa57dbf89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.864ex; height:3.176ex;" alt="{\displaystyle H^{2}(\mathbb {R} ^{n})}" loading="lazy"> der zweimal schwach differenzierbaren quadratische integrierbaren Funktionen ist ein stetiger gleichmäßig elliptischer Operator. Daher ist er auch ein Fredholm-Operator. Da er auch <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungiert</a> ist, hat er den Fredholm-Index 0.

Betrachtet man den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> im <a href="Distributionelle_Ableitung" class="mw-redirect" title="Distributionelle Ableitung">distributionellen Sinn</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f15dffc817bd5f8767061d98adb57caf3519f46f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{n})}" loading="lazy">, ist er kein stetiger Operator und somit kein Fredholm-Operator bezüglich der obigen Definition. Im Sinne von unbeschränkten Operatoren, wie dies später im Artikel noch erklärt wird, ist er allerdings weiterhin ein Fredholm-Operator.

<div class="mw-heading mw-heading3"><h3 id="Elliptische_Operatoren">Elliptische Operatoren</h3></div>

Jeder <a href="Elliptischer_Differentialoperator" class="mw-redirect" title="Elliptischer Differentialoperator">gleichmäßig elliptische Differentialoperator</a> ist ein Fredholm-Operator.

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79774d994aac0be34ef390915fed12cbce816f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.673ex; height:2.343ex;" alt="{\displaystyle \Omega \subset \mathbb {R} ^{n}}" loading="lazy"> ein <a href="Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> mit <a href="Lipschitz-Rand" class="mw-redirect" title="Lipschitz-Rand">Lipschitz-Rand</a>. Dann ist der <a href="Schwache_Ableitung" title="Schwache Ableitung">schwache</a>, elliptische Differentialoperator mit homogenen <a href="Neumann-Randbedingung" title="Neumann-Randbedingung">Neumann-Randbedingungen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon H^{1,2}(\Omega )\to H^{1,2}(\Omega )'}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon H^{1,2}(\Omega )\to H^{1,2}(\Omega )'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1d732924f5633d76808aafc4cf46af2df5042df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.925ex; height:3.176ex;" alt="{\displaystyle A\colon H^{1,2}(\Omega )\to H^{1,2}(\Omega )'}" loading="lazy"> definiert durch

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(u)(v):=\int _{\Omega }\sum _{i,j}\partial _{i}va_{ij}\partial _{j}u}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1de5b104502ed568a533949882f86a7a11ab07b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.747ex; height:6.676ex;" alt="{\displaystyle A(u)(v):=\int _{\Omega }\sum _{i,j}\partial _{i}va_{ij}\partial _{j}u}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in H^{1,2}(\Omega )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v\in H^{1,2}(\Omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f9660f44d09ba5cf0ae11062037c4c9b9eb99c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.256ex; height:3.176ex;" alt="{\displaystyle u,v\in H^{1,2}(\Omega )}" loading="lazy"> ein Fredholm-Operator.

<div class="mw-heading mw-heading3"><h3 id="Elliptischer_Operator_auf_einer_Mannigfaltigkeit">Elliptischer Operator auf einer Mannigfaltigkeit</h3></div>

Der <a href="Kreis" title="Kreis">Kreis</a> (als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} /\mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f62160dc28538c41d74ffcfbca4a7b3c68693880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.391ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} /\mathbb {Z} }" loading="lazy"> gedacht) kann als eindimensionale <a href="Geschlossene_Mannigfaltigkeit" title="Geschlossene Mannigfaltigkeit">geschlossene Mannigfaltigkeit</a> verstanden werden. Ein stetiger elliptischer Differentialoperator erster Ordnung auf den glatten Funktionen vom Kreis in die komplexen Zahlen ist durch

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D={\frac {\mathrm {d} }{\mathrm {d} x}}-2\pi i\lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f183317ab4e98f0c894987051fb178363dcbe2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.974ex; height:5.509ex;" alt="{\displaystyle D={\frac {\mathrm {d} }{\mathrm {d} x}}-2\pi i\lambda }" loading="lazy"></dd></dl>

für eine komplexe Konstante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> gegeben. Der Kern von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> ist der von den Termen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(i2\pi \,\lambda \,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(i2\pi \,\lambda \,x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6e900994eba0d6a86653a36a0859e13d76dd997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.118ex; height:2.843ex;" alt="{\displaystyle \exp(i2\pi \,\lambda \,x)}" loading="lazy"> aufgespannte Raum, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bbb4fa78e592bed8b3dc2dbb47888e375f66511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.746ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {Z} }" loading="lazy">, und 0 in den anderen Fällen. Der Kern des adjungierten Operators ist ein ähnlicher Raum, nur wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> durch sein komplex-konjugiertes ersetzt. Der Fredholm-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> hat damit den Index 0. Dieses Beispiel zeigt, dass Kern und Kokern eines elliptischen Operators unstetig springen können, falls man den elliptischen Operator so variiert, dass die oben erwähnten Terme erfasst werden. Da die Sprünge in den Dimensionen von Kern und Kokern aber gleich sind, ändert sich ihre Differenz, der Index, stetig.

<div class="mw-heading mw-heading2"><h2 id="Unbeschränkte_Fredholm-Operatoren">Unbeschränkte Fredholm-Operatoren</h2></div>

Bisher wurden in diesem Artikel Fredholm-Operatoren nur als spezielle beschränkte Operatoren betrachtet. Beispielsweise in der Indextheorie elliptischer Operatoren über nicht kompakten Räumen ist es jedoch sinnvoll die Definition des Fredholm-Operators auf unbeschränkte Operatoren zu erweitern. Die Definition ist bis auf die geforderte Abgeschlossenheit des Operators identisch mit der im beschränkten Fall:

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> zwei Banachräume und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}(A)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d2e410bd398eb0ec1fa11c6d14aead65b499fe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.344ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(A)}" loading="lazy"> ein Unterraum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Ein (unbeschränkter) Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon X\supset {\mathcal {D}}(A)\to Y}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\colon X\supset {\mathcal {D}}(A)\to Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30e7f3630316715328b556d37091e8c0551db0d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.587ex; height:2.843ex;" alt="{\displaystyle A\colon X\supset {\mathcal {D}}(A)\to Y}" loading="lazy"> wird Fredholm-Operator genannt, falls

</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"> <a href="Abgeschlossener_Operator" title="Abgeschlossener Operator">abgeschlossen</a> ist,</li>

<li>die Dimension des Kerns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ker} (A)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fab27fb4326a4df502f995954130806bc0311ecb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.724ex; height:2.843ex;" alt="{\displaystyle \operatorname {ker} (A)}" loading="lazy"> endlich ist,</li>

<li>die Kodimension von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ran} (A)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2661b1fb847ee5e1eb423afae01ad2288ee564d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.919ex; height:2.843ex;" alt="{\displaystyle \operatorname {ran} (A)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> endlich ist.</li></ul>

Manche Autoren verlangen zusätzlich, dass der Definitionsbereich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}(A)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d2e410bd398eb0ec1fa11c6d14aead65b499fe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.344ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(A)}" loading="lazy"> dicht liegt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, was aber offensichtlich völlig unabhängig von der eigentlichen Fredholm-Eigenschaft ist. Der Fredholm-Index ist wie im Fall beschränkter Operatoren durch

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ind} (A)=\dim(\ker A)-\mathrm {codim} (\mathrm {ran} (A))\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/898a5b5156ba78fe33763e87fe7a0ad4f6c1757e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.899ex; height:2.843ex;" alt="{\displaystyle \mathrm {ind} (A)=\dim(\ker A)-\mathrm {codim} (\mathrm {ran} (A))\in \mathbb {Z} }" loading="lazy"></dd></dl>

definiert.

Versieht man den Definitionsbereich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}(A)\subset X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}(A)\subset X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5b1870fd3c4545c2185b929c53be705c7ada354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.423ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(A)\subset X}" loading="lazy"> eines abgeschlossenen Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3fcb04477030783ab411037da34caeec858b0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.144ex; height:2.176ex;" alt="{\displaystyle A\colon X\to Y}" loading="lazy"> mit der sogenannten Graphennorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{A}:=\|x\|_{X}+\|Ax\|_{Y}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|x\|_{A}:=\|x\|_{X}+\|Ax\|_{Y}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/897fecd83c34493d6f7bdb026e97f5b43c6821b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.876ex; height:2.843ex;" alt="{\displaystyle \|x\|_{A}:=\|x\|_{X}+\|Ax\|_{Y}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}:=({\mathcal {D}}(A),\|\cdot \|_{A})}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{A}:=({\mathcal {D}}(A),\|\cdot \|_{A})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45a19979bf35733b49fa19094a00c65631ea9dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.791ex; height:2.843ex;" alt="{\displaystyle X_{A}:=({\mathcal {D}}(A),\|\cdot \|_{A})}" loading="lazy"> ein Banachraum und <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">, betrachtet als Operator von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, ein beschränkter Operator. Folglich kann ein unbeschränkter Fredholm-Operator stets auf einen beschränkten Fredholm-Operator zurückgeführt werden. Dementsprechend gelten viele Eigenschaften von oben auch für unbeschränkte Fredholm-Operatoren. So ist die Verkettung unbeschränkter Fredholm-Operatoren wieder ein Fredholm-Operator, für den obige Indexformel gilt; der Satz von Atkinson gilt ebenfalls, und der Fredholm-Index unbeschränkter Fredholm-Operatoren ist auch invariant unter kompakten Störungen und lokal konstant (das Wort "lokal" bezieht sich hierbei auf die so genannte Gap-Metrik). Schließlich gilt auch das Punctured Neighborhood Theorem für unbeschränkte Fredholm-Operatoren. Eine Verbindung zur Calkin-Algebra besteht für unbeschränkte Fredholm-Operatoren allerdings nicht.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Wesentliches_Spektrum" title="Wesentliches Spektrum">Wesentliches Spektrum</a></li></ul>

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

<ul><li><a href="Bernhard_Schiekel" title="Bernhard Schiekel">Bernhard Schiekel</a>: Krümmungen und Indexsätze – auf den Spuren von Gauß-Bonnet, Cartan, Atiyah-Singer und Witten. Eine Einführung in Geometrie und Topologie für Physiker. 2. Aufl. <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.18725/OPARU-17162">10.18725/OPARU-17162</a></li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Noetherian_operator">Noether operator</a> in Encyclopdia of Mathematics</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> Fritz Noether: <cite style="font-style:italic">Über eine Klasse singulärer Integralgleichungen</cite>. In: <cite style="font-style:italic">Mathematische Annalen</cite>. Band 82, Nr. 1–2, März 1920, <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%220025-5831%22&key=cql">0025-5831</a>, S. 42–63, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/bf01457974">10.1007/bf01457974</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.atitle=%C3%9Cber+eine+Klasse+singul%C3%A4rer+Integralgleichungen&rft.au=Fritz+Noether&rft.date=1920-03&rft.doi=10.1007%2Fbf01457974&rft.genre=journal&rft.issn=0025-5831&rft.issue=1-2&rft.jtitle=Mathematische+Annalen&rft.pages=42-63&rft.volume=82" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> V. G. Maz'ya: <cite style="font-style:italic">Analysis IV: Linear and Boundary Integral Equations</cite>. Springer Science & Business Media, 2012, ISBN 978-3-642-58175-5, S. 24–25 (<a rel="nofollow" class="external text" href="https://www.google.de/books/edition/Analysis_IV/AWPyCAAAQBAJ?hl=de&gbpv=1&dq=lang+noether+operator&pg=PA23&printsec=frontcover">google.de</a> [abgerufen am 16. Januar 2025]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=V.+G.+Maz%27ya&rft.btitle=Analysis+IV%3A+Linear+and+Boundary+Integral+Equations&rft.date=2012-12-06&rft.genre=book&rft.isbn=9783642581755&rft.pages=24-25&rft.pub=Springer+Science+%26+Business+Media" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Dirk Werner: <cite style="font-style:italic">Funktionalanalysis</cite>. Springer-Verlag, 2007, ISBN 978-3-540-72536-7, S. 261, 483 f. (<a rel="nofollow" class="external text" href="https://www.google.de/books/edition/Funktionalanalysis/e7gfBAAAQBAJ?hl=de&gbpv=1&dq=werner+fredholm-operator&pg=PA484&printsec=frontcover">google.de</a> [abgerufen am 16. Januar 2025]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=Dirk+Werner&rft.btitle=Funktionalanalysis&rft.date=2007-08-24&rft.genre=book&rft.isbn=9783540725367&rft.pages=261%2C+483f&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Masoud Khalkhali: <cite style="font-style:italic">Basic Noncommutative Geometry</cite>. 2. Auflage. EMS, 2013, ISBN 978-3-03719-128-6, S. 201.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=Masoud+Khalkhali&rft.btitle=Basic+Noncommutative+Geometry&rft.date=2013&rft.edition=2.&rft.genre=book&rft.isbn=9783037191286&rft.pages=201&rft.pub=EMS" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Jürgen Appell, <a href="Martin_V%C3%A4th" title="Martin Väth">Martin Väth</a>: <cite style="font-style:italic">Elemente der Funktionalanalysis</cite>. Springer/Vieweg, 2005, ISBN 3-322-80243-4, S. 164–165.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=J%C3%BCrgen+Appell%2C+Martin+V%C3%A4th&rft.btitle=Elemente+der+Funktionalanalysis&rft.date=2005&rft.genre=book&rft.isbn=3322802434&rft.pages=164-165&rft.pub=Springer%2FVieweg" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Vladimir Müller: <cite style="font-style:italic">Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras</cite>. Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0, S. 159.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=Vladimir+M%C3%BCller&rft.btitle=Spectral+Theory+of+Linear+Operators%3A+and+Spectral+Systems+in+Banach+Algebras&rft.date=2007&rft.genre=book&rft.isbn=9783764382650&rft.pages=159&rft.place=Basel&rft.pub=Birkh%C3%A4user" style="display:none">

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Vladimir Müller: <cite style="font-style:italic">Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras</cite>. Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0, S. 156.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=Vladimir+M%C3%BCller&rft.btitle=Spectral+Theory+of+Linear+Operators%3A+and+Spectral+Systems+in+Banach+Algebras&rft.date=2007&rft.genre=book&rft.isbn=9783764382650&rft.pages=156&rft.place=Basel&rft.pub=Birkh%C3%A4user" style="display:none">

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Vladimir Müller: <cite style="font-style:italic">Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras</cite>. Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0, S. 171, 293–294.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=Vladimir+M%C3%BCller&rft.btitle=Spectral+Theory+of+Linear+Operators%3A+and+Spectral+Systems+in+Banach+Algebras&rft.date=2007&rft.genre=book&rft.isbn=9783764382650&rft.pages=171%2C+293-294&rft.place=Basel&rft.pub=Birkh%C3%A4user" style="display:none">

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Vladimir Müller: <cite style="font-style:italic">Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras</cite>. Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0, S. 231.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fredholm-Operator&rft.au=Vladimir+M%C3%BCller&rft.btitle=Spectral+Theory+of+Linear+Operators%3A+and+Spectral+Systems+in+Banach+Algebras&rft.date=2007&rft.genre=book&rft.isbn=9783764382650&rft.pages=231&rft.place=Basel&rft.pub=Birkh%C3%A4user" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Martin Schechter: Fredholm Operators and the Essential Spectrum. <a rel="nofollow" class="external text" href="https://archive.org/details/fredholmoperator00sche">(online)</a>

</li>

</ol>

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Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4155256-8">4155256-8</a> | <a href="Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85051649">sh85051649</a> </div>

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