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<title>Unitärer Operator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Unitärer Operator</span></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"><p>Ein <b>unitärer Operator</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> ein <a href="Bijektive_Funktion" title="Bijektive Funktion">bijektiver</a> <a href="Linearer_Operator" title="Linearer Operator">linearer Operator</a> zwischen zwei <a href="Hilbertraum" title="Hilbertraum">Hilberträumen</a>, der das <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> erhält. Unitäre Operatoren sind damit spezielle <a href="Orthogonale_Abbildung" title="Orthogonale Abbildung">orthogonale</a> oder <a href="Unit%C3%A4re_Abbildung" title="Unitäre Abbildung">unitäre Abbildungen</a> und stets <a href="Norm_(Mathematik)" title="Norm (Mathematik)">normerhaltend</a>, <a href="Isometrie" title="Isometrie">abstandserhaltend</a>, <a href="Beschr%C3%A4nkter_Operator" title="Beschränkter Operator">beschränkt</a> und, falls beide Hilberträume gleich sind, <a href="Normaler_Operator" title="Normaler Operator">normal</a>. Der <a href="Inverse_Funktion" class="mw-redirect" title="Inverse Funktion">inverse Operator</a> eines unitären Operators ist gleich seinem <a href="Adjungierter_Operator" title="Adjungierter Operator">adjungierten Operator</a>. Die <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> eines unitären Operators in einem Hilbertraum haben alle den <a href="Betragsfunktion" title="Betragsfunktion">Betrag</a> eins. Unitäre Operatoren zwischen endlichdimensionalen Vektorräumen gleicher Dimension können nach Wahl je einer <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> durch <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitäre Matrizen</a> dargestellt werden. Wichtige Beispiele für unitäre Operatoren zwischen unendlichdimensionalen <a href="Funktionenraum" title="Funktionenraum">Funktionenräumen</a> sind die <a href="Fouriertransformation" class="mw-redirect" title="Fouriertransformation">Fouriertransformation</a> und die <a href="Zeitentwicklungsoperator" title="Zeitentwicklungsoperator">Zeitentwicklungsoperatoren</a> der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a>.

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

<p>Ein unitärer Operator ist ein <a href="Bijektive_Funktion" title="Bijektive Funktion">bijektiver</a> <a href="Linearer_Operator" title="Linearer Operator">linearer Operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to W}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9307c45fa2557b519d013c5b4229c9bd53a6c3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.176ex;" alt="{\displaystyle T\colon V\to W}" loading="lazy"></span> zwischen zwei <a href="Hilbertraum" title="Hilbertraum">Hilberträumen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\langle \cdot ,\cdot \rangle _{V})}">

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<annotation encoding="application/x-tex">{\displaystyle (V,\langle \cdot ,\cdot \rangle _{V})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e0c8d79c1a5cb5801fea57a9fefd2aa0732279.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.263ex; height:2.843ex;" alt="{\displaystyle (V,\langle \cdot ,\cdot \rangle _{V})}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W,\langle \cdot ,\cdot \rangle _{W})}">

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<annotation encoding="application/x-tex">{\displaystyle (W,\langle \cdot ,\cdot \rangle _{W})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfa690494895a3ef42e43d5515add29f5b0f0056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.37ex; height:2.843ex;" alt="{\displaystyle (W,\langle \cdot ,\cdot \rangle _{W})}" loading="lazy"></span>, sodass

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<annotation encoding="application/x-tex">{\displaystyle \langle Tu,Tv\rangle _{W}=\langle u,v\rangle _{V}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a11c78859410db25dcaf2c3141d8300ba8813ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.422ex; height:2.843ex;" alt="{\displaystyle \langle Tu,Tv\rangle _{W}=\langle u,v\rangle _{V}}" loading="lazy"></span></dd></dl>

<p>für alle Vektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in V}">

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<annotation encoding="application/x-tex">{\displaystyle u,v\in V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7acb96be1087c3c2f30d303aa4cac24f62f45daf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.119ex; height:2.509ex;" alt="{\displaystyle u,v\in V}" loading="lazy"></span> gilt. Ein unitärer Operator ist demnach ein <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a> zwischen zwei Hilberträumen, der das <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> erhält. Ein unitärer Operator zwischen zwei reellen Hilberträumen wird gelegentlich auch als <i>orthogonaler Operator</i> bezeichnet.

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

<p>Im Folgenden werden die Zusätze <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,W}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a40b0deabeee6e15bff1e3079b601986d8fe337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.256ex; height:2.509ex;" alt="{\displaystyle V,W}" loading="lazy"></span> bei den Skalarprodukten weggelassen, da durch das Argument klar wird, um welchen Raum es sich jeweils handelt.

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

<p>Jeder unitäre Operator stellt eine <a href="Unit%C3%A4re_Abbildung" title="Unitäre Abbildung">unitäre Abbildung</a> (im reellen Fall <a href="Orthogonale_Abbildung" title="Orthogonale Abbildung">orthogonale Abbildung</a>) dar. Die <a href="Lineare_Abbildung" title="Lineare Abbildung">Linearität</a> folgt daher bereits aus der Erhaltung des Skalarprodukts und muss demnach nicht separat gefordert werden. Ein unitärer Operator erhält weiterhin die <a href="Skalarproduktnorm" title="Skalarproduktnorm">Skalarproduktnorm</a> eines Vektors, das heißt, es gilt

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<annotation encoding="application/x-tex">{\displaystyle \|Tv\|={\sqrt {\langle Tv,Tv\rangle }}={\sqrt {\langle v,v\rangle }}=\|v\|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3ba80efb8981b458da6fb93a59cfec788400bdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.954ex; height:4.843ex;" alt="{\displaystyle \|Tv\|={\sqrt {\langle Tv,Tv\rangle }}={\sqrt {\langle v,v\rangle }}=\|v\|}" loading="lazy"></span>,</dd></dl>

<p>und damit auch den <a href="Abstand" title="Abstand">Abstand</a> zweier Vektoren. Die Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math></span><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"></span> stellt somit eine <a href="Isometrie" title="Isometrie">Isometrie</a> dar und die beiden Räume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math></span><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"></span> sind daher <a href="Isometrische_Isomorphie" title="Isometrische Isomorphie">isometrisch isomorph</a>. Die <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> eines unitären Operators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3a6ee7fc18a25eec79f8b0e2adff07caea23a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.859ex; height:2.176ex;" alt="{\displaystyle T\colon V\to V}" loading="lazy"></span> haben alle den <a href="Betragsfunktion" title="Betragsfunktion">Betrag</a> eins. Allgemeiner liegt das <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrum</a> eines unitären Operators im Rand des <a href="Einheitskreis" title="Einheitskreis">Einheitskreises</a>.

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

<p>Für die <a href="Operatornorm" title="Operatornorm">Operatornorm</a> eines unitären Operators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math></span><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"></span> gilt aufgrund der Normerhaltung

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<annotation encoding="application/x-tex">{\displaystyle \|T\|=\sup _{\|v\|=1}\|Tv\|=\sup _{\|v\|=1}\|v\|=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6885b7223e3718182d070966f1ce762b19421663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.818ex; height:5.009ex;" alt="{\displaystyle \|T\|=\sup _{\|v\|=1}\|Tv\|=\sup _{\|v\|=1}\|v\|=1}" loading="lazy"></span>.</dd></dl>

<p>Ein unitärer Operator ist demnach immer <a href="Beschr%C3%A4nkter_Operator" title="Beschränkter Operator">beschränkt</a> und damit <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a>.

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

<p>Der <a href="Inverse_Abbildung" class="mw-redirect" title="Inverse Abbildung">inverse Operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{-1}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a1a5bed0ce2d8adc7fdad412c34ef905fb5026f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.053ex; height:2.676ex;" alt="{\displaystyle T^{-1}}" loading="lazy"></span> eines unitären Operators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math></span><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"></span> ist gleich seinem <a href="Adjungierter_Operator" title="Adjungierter Operator">adjungierten Operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\ast }}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fa50cc66f556e4cec7fa230181a9994dcab6672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.343ex;" alt="{\displaystyle T^{\ast }}" loading="lazy"></span>, also

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4960fe7d65956b57fdb9d67c85701901dafcd1ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.925ex; height:2.676ex;" alt="{\displaystyle T^{-1}=T^{\ast }}" loading="lazy"></span>,</dd></dl>

<p>denn es gilt

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<annotation encoding="application/x-tex">{\displaystyle \langle u,T^{\ast }v\rangle =\langle Tu,v\rangle =\langle Tu,TT^{-1}v\rangle =\langle u,T^{-1}v\rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2877720fe321559d6fb07ecedbfcbfe6d1c03307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.286ex; height:3.176ex;" alt="{\displaystyle \langle u,T^{\ast }v\rangle =\langle Tu,v\rangle =\langle Tu,TT^{-1}v\rangle =\langle u,T^{-1}v\rangle }" loading="lazy"></span>.</dd></dl>

<p>Stimmen umgekehrt Inverse und Adjungierte eines linearen Operators überein, dann ist dieser unitär, denn es gilt

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<annotation encoding="application/x-tex">{\displaystyle \langle Tu,Tv\rangle =\langle u,T^{\ast }Tv\rangle =\langle u,T^{-1}Tv\rangle =\langle u,v\rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a8cf3c9865265626370d853f73aad6f7d3eff74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.87ex; height:3.176ex;" alt="{\displaystyle \langle Tu,Tv\rangle =\langle u,T^{\ast }Tv\rangle =\langle u,T^{-1}Tv\rangle =\langle u,v\rangle }" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Normalität"><span id="Normalit.C3.A4t"></span>Normalität</h3></div>

<p>Aufgrund der Übereinstimmung von Inverser und Adjungierter ist ein unitärer Operator im Fall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=W}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/740038d36bd79466d6938d73b83fe737161fa1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle V=W}" loading="lazy"></span> stets <a href="Normaler_Operator" title="Normaler Operator">normal</a>, das heißt

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eca360f1f788996662efb28e8152e0ad2e8d9de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.19ex; height:2.343ex;" alt="{\displaystyle T^{\ast }T=TT^{\ast }=I}" loading="lazy"></span>.</dd></dl>

<p>Für unitäre Operatoren auf komplexen Hilberträumen und <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungierte</a> unitäre Operatoren auf reellen Hilberträumen gilt damit der <a href="Spektralsatz" title="Spektralsatz">Spektralsatz</a>.

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

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

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</math></span><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"></span> ein unitärer Operator und ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{i})_{i\in I}}">

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<annotation encoding="application/x-tex">{\displaystyle (v_{i})_{i\in I}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a41d171710e2f4ea57c79d3ff95ec971e2bed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle (v_{i})_{i\in I}}" loading="lazy"></span> eine <a href="Hilbertbasis" class="mw-redirect" title="Hilbertbasis">Hilbertbasis</a> (ein vollständiges Orthonormalsystem) von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Tv_{i})_{i\in I}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617fce52151d1a55b6981728f049ef0a90be1c3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.097ex; height:2.843ex;" alt="{\displaystyle (Tv_{i})_{i\in I}}" loading="lazy"></span> eine Hilbertbasis von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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</math></span><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"></span>, denn es gilt

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<annotation encoding="application/x-tex">{\displaystyle \langle Tv_{i},Tv_{j}\rangle =\langle v_{i},v_{j}\rangle =\delta _{ij}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f33f03e760446e87c7ccf8e7d90b890fd3b90382.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.595ex; height:3.009ex;" alt="{\displaystyle \langle Tv_{i},Tv_{j}\rangle =\langle v_{i},v_{j}\rangle =\delta _{ij}}" loading="lazy"></span>.</dd></dl>

<p>Sind umgekehrt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{i})_{i\in I}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a41d171710e2f4ea57c79d3ff95ec971e2bed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle (v_{i})_{i\in I}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Tv_{i})_{i\in I}}">

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<annotation encoding="application/x-tex">{\displaystyle (Tv_{i})_{i\in I}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617fce52151d1a55b6981728f049ef0a90be1c3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.097ex; height:2.843ex;" alt="{\displaystyle (Tv_{i})_{i\in I}}" loading="lazy"></span> Hilbertbasen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math></span><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"></span> und ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math></span><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"></span> linear, so folgt daraus die Unitarität von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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</math></span><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"></span>, denn man erhält

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle Tu,Tv\rangle &={\big \langle }T{\big (}{\textstyle \sum _{i}}\lambda _{i}v_{i}{\big )},T{\big (}{\textstyle \sum _{j}}\mu _{j}v_{j}{\big )}{\big \rangle }={\big \langle }{\textstyle \sum _{i}}\lambda _{i}Tv_{i},{\textstyle \sum _{j}}\mu _{j}Tv_{j}{\big \rangle }={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}{\big \langle }Tv_{i},Tv_{j}{\big \rangle }=\\&={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}\delta _{ij}={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}\langle v_{i},v_{j}\rangle ={\big \langle }{\textstyle \sum _{i}}\lambda _{i}v_{i},{\textstyle \sum _{j}}\mu _{j}v_{j}{\big \rangle }=\langle u,v\rangle .\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc296cd328e519e61564ddcac848cbcb8ce60520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:89.981ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\langle Tu,Tv\rangle &={\big \langle }T{\big (}{\textstyle \sum _{i}}\lambda _{i}v_{i}{\big )},T{\big (}{\textstyle \sum _{j}}\mu _{j}v_{j}{\big )}{\big \rangle }={\big \langle }{\textstyle \sum _{i}}\lambda _{i}Tv_{i},{\textstyle \sum _{j}}\mu _{j}Tv_{j}{\big \rangle }={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}{\big \langle }Tv_{i},Tv_{j}{\big \rangle }=\\&={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}\delta _{ij}={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}\langle v_{i},v_{j}\rangle ={\big \langle }{\textstyle \sum _{i}}\lambda _{i}v_{i},{\textstyle \sum _{j}}\mu _{j}v_{j}{\big \rangle }=\langle u,v\rangle .\end{aligned}}}" loading="lazy"></span></dd></dl>

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

<ul><li><a href="Hilbert-Schmidt-Operator" title="Hilbert-Schmidt-Operator">Hilbert-Schmidt-Operator</a></li>

<li><a href="Hilbertraum-Darstellung" title="Hilbertraum-Darstellung">Hilbertraum-Darstellung</a></li></ul>

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

<ul><li>Hans Wilhelm Alt: <cite style="font-style:italic">Lineare Funktionalanalysis: Eine anwendungsorientierte Einführung</cite>. 5. Auflage. Springer, 2008, ISBN 3-540-34186-2.<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:Unit%C3%A4rer+Operator&rft.au=Hans+Wilhelm+Alt&rft.btitle=Lineare+Funktionalanalysis%3A+Eine+anwendungsorientierte+Einf%C3%BChrung&rft.date=2008&rft.edition=5.&rft.genre=book&rft.isbn=3540341862&rft.pub=Springer" style="display:none"> </span></li>

<li>Dirk Werner: <cite style="font-style:italic">Funktionalanalysis</cite>. 5. Auflage. Springer, 2005, ISBN 3-540-21381-3.<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:Unit%C3%A4rer+Operator&rft.au=Dirk+Werner&rft.btitle=Funktionalanalysis&rft.date=2005&rft.edition=5.&rft.genre=book&rft.isbn=3540213813&rft.pub=Springer" style="display:none"> </span></li></ul>

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

<ul><li>V.I. Sobolev: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Unitary operator</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Unitary_operator">encyclopediaofmath.org</a>).<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:Unit%C3%A4rer+Operator&rft.atitle=Unitary+operator&rft.au=V.I.+Sobolev&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li>

<li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Unitary.html"><i>Unitary</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>

<li>asteroid: <a rel="nofollow" class="external text" href="https://planetmath.org/unitary"><i>Unitary</i>.</a> In: <i><a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</i> (englisch)</li></ul></div><div><div class="zim-footer">

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