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<title>Normaler Operator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Normaler 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>In der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a> verallgemeinert der <b>normale Operator</b> den Begriff der <a href="Normale_Matrix" title="Normale Matrix">normalen Matrix</a> aus der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>.

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

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

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</math></span><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"></span> ein <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> und bezeichnet <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 {\mathcal {L}}(X)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5ca7f93cc552ef4dd32f4f56f0fd3fb5971107c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.393ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(X)}" loading="lazy"></span> die Menge aller <a href="Beschr%C3%A4nkter_Operator" title="Beschränkter Operator">stetigen</a> <a href="Endomorphismus" title="Endomorphismus">Endomorphismen</a> 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 X}">

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

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</math></span><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"></span>, so heißt ein Operator <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 A\in {\mathcal {L}}(X)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32f4625104c3b6211705e13f53eed262c9924424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.977ex; height:2.843ex;" alt="{\displaystyle A\in {\mathcal {L}}(X)}" loading="lazy"></span> normal, falls er mit 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 A^{\ast }}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5541bfa07743be995242c892c344395e672d6fa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{\ast }}" loading="lazy"></span> kommutiert, also wenn

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62660667dd8d36fe320fe23e24fb74f52228d12d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.179ex; height:2.343ex;" alt="{\displaystyle AA^{\ast }=A^{\ast }A}" loading="lazy"></span></dd></dl>

<p>gilt.

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

<ul><li><a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">Selbstadjungierte</a> und <a href="Unit%C3%A4rer_Operator" title="Unitärer Operator">unitäre</a> Operatoren sind offenbar normal.</li>

<li>Der <a href="Shift-Operator" class="mw-redirect" title="Shift-Operator">unilaterale Shift</a> ist ein Beispiel für einen nicht-normalen Operator.</li></ul>

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

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32f4625104c3b6211705e13f53eed262c9924424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.977ex; height:2.843ex;" alt="{\displaystyle A\in {\mathcal {L}}(X)}" loading="lazy"></span> ein normaler Operator. Dann gilt:

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/596c85a7d7cc822c18754bc0f5d475f7489b94a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.948ex; height:2.843ex;" alt="{\displaystyle \|Ax\|=\|A^{\ast }x\|}" loading="lazy"></span> für alle <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 x\in X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f77332619cba2c0871f68b2be209d7d6717ee325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.657ex; height:3.176ex;" alt="{\displaystyle \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}" loading="lazy"></span> für alle <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 x\in X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span></li>

<li>Die <a href="Operatornorm" title="Operatornorm">Operatornorm</a> 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 A}">

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

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</math></span><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"></span> ist gleich dem <a href="Spektralradius" title="Spektralradius">Spektralradius</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 \|A\|=\sup\{|\lambda |\colon \lambda \in \sigma (A)\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \|A\|=\sup\{|\lambda |\colon \lambda \in \sigma (A)\}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b914ba7eed7a6130de7b25de57f2859147c4f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.4ex; height:2.843ex;" alt="{\displaystyle \|A\|=\sup\{|\lambda |\colon \lambda \in \sigma (A)\}.}" loading="lazy"></span> Dabei bezeichnet <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 \sigma (A)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"></span> das <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrum</a> 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 A}">

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

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</math></span><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"></span>.</li>

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

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</math></span><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"></span> erzeugte <a href="C*-Algebra" title="C*-Algebra">C<sup>*</sup>-Algebra</a> und die 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 A}">

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

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</math></span><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"></span> erzeugte <a href="Von-Neumann-Algebra" title="Von-Neumann-Algebra">Von-Neumann-Algebra</a> sind kommutativ. Dieser Sachverhalt ermöglicht einen <a href="Funktionalkalk%C3%BCl" title="Funktionalkalkül">Funktionalkalkül</a>.</li>

<li>Die <a href="Diagonalisierbarkeit" class="mw-redirect" title="Diagonalisierbarkeit">Diagonalisierbarkeit</a> normaler Matrizen in der linearen Algebra verallgemeinert sich auf normale Operatoren in Form des <a href="Spektralsatz" title="Spektralsatz">Spektralsatzes</a>.</li>

<li>Eine Klassifikation normaler Operatoren besteht bzgl. <a href="Unit%C3%A4re_Abbildung" title="Unitäre Abbildung">unitärer</a> Äquivalenz modulo <a href="Kompakter_Operator" title="Kompakter Operator">kompakter Operatoren</a>, indem man zur <a href="Calkin-Algebra" title="Calkin-Algebra">Calkin-Algebra</a> übergeht, die im endlich-dimensionalen 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 \{0\}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"></span> ist. Das ist im Artikel zur Calkin-Algebra ausgeführt.</li>

<li>Ein beschränkter Operator <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 A}">

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

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</math></span><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"></span> in einem komplexen Hilbertraum lässt sich zerlegen in <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 A=W_{1}+i\,W_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A=W_{1}+i\,W_{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/577b1bbcba2aeafc37913ed6095ae5346ed7e815.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.367ex; height:2.509ex;" alt="{\displaystyle A=W_{1}+i\,W_{2}}" loading="lazy"></span> mit dem „Realteil“ <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_{1}={\tfrac {1}{2}}(A+A^{\ast })}">

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<annotation encoding="application/x-tex">{\displaystyle W_{1}={\tfrac {1}{2}}(A+A^{\ast })}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31410b7cdfcdb66acddf8ff2178a278052db641b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.195ex; height:3.509ex;" alt="{\displaystyle W_{1}={\tfrac {1}{2}}(A+A^{\ast })}" loading="lazy"></span> und dem „Imaginärteil“ <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_{2}={\tfrac {1}{2i}}(A-A^{\ast }).}">

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<annotation encoding="application/x-tex">{\displaystyle W_{2}={\tfrac {1}{2i}}(A-A^{\ast }).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5508145d2df2acaf1f48a0573e58fb737c9c368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.409ex; height:3.509ex;" alt="{\displaystyle W_{2}={\tfrac {1}{2i}}(A-A^{\ast }).}" loading="lazy"></span> Dabei sind die Operatoren <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_{i}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}" loading="lazy"></span> <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungiert</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 A}">

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

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</math></span><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"></span> ist genau dann normal, wenn <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_{1}W_{2}=W_{2}W_{1}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50414e28c9248963f839a58a1838ac3199be986f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.09ex; height:2.509ex;" alt="{\displaystyle W_{1}W_{2}=W_{2}W_{1}}" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Verwandte_Begriffe">Verwandte Begriffe</h2></div>

<p>Ein Operator <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 A\in {\mathcal {L}}(X)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32f4625104c3b6211705e13f53eed262c9924424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.977ex; height:2.843ex;" alt="{\displaystyle A\in {\mathcal {L}}(X)}" loading="lazy"></span> heißt

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<ul><li><i>quasinormal</i>, falls <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 A\,\!}">

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b041434092d1f0042c2b4c7ab32ea84d462cb53e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}" loading="lazy"></span> mit <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 A^{\ast }A}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a23dc1b6340d70dcf224080f8e5da82a5d8534cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.54ex; height:2.343ex;" alt="{\displaystyle A^{\ast }A}" loading="lazy"></span> vertauscht, das heißt <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 AA^{\ast }A=A^{\ast }AA}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3819b2e0d77d994090394a67c29537fe621d818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.666ex; height:2.343ex;" alt="{\displaystyle AA^{\ast }A=A^{\ast }AA}" loading="lazy"></span>.</li>

<li><i>subnormal</i>, falls es einen Hilbertraum <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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> gibt, so dass <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 X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> Unterraum 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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> ist, und einen normalen Operator <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 B\in {\mathcal {L}}(Y)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/652e1c784c560e0a624e9bef02f0b02ad586841f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.791ex; height:2.843ex;" alt="{\displaystyle B\in {\mathcal {L}}(Y)}" loading="lazy"></span>, so dass <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 B(X)\subset X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b98220e0b68232bb188ee60a214af919dcd2abb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.632ex; height:2.843ex;" alt="{\displaystyle B(X)\subset X}" 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 A=B|_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6ff9d53047af720d18323d91e4c2022a5b88c98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.885ex; height:2.843ex;" alt="{\displaystyle A=B|_{X}}" loading="lazy"></span>.</li>

<li><i>hyponormal</i>, falls <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 \|A^{\ast }x\|\leq \|Ax\|}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|A^{\ast }x\|\leq \|Ax\|}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/521225d5443acf5ee7de72db08d0d6dd9ed02616.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.948ex; height:2.843ex;" alt="{\displaystyle \|A^{\ast }x\|\leq \|Ax\|}" loading="lazy"></span> für alle <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 x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>.</li>

<li><i>paranormal</i>, falls <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 \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f77332619cba2c0871f68b2be209d7d6717ee325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.657ex; height:3.176ex;" alt="{\displaystyle \|Ax\|^{2}\leq \|A^{2}x\|\|x\|}" loading="lazy"></span> für alle <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 x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>.</li>

<li><i>normaloid</i>, falls Operatornorm = Spektralradius, d. h.: <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 \|A\|=\sup\{|\lambda |;\lambda \in \sigma (A)\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|A\|=\sup\{|\lambda |;\lambda \in \sigma (A)\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2791a3d2d51b59abb312340386582d484f9b54d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.754ex; height:2.843ex;" alt="{\displaystyle \|A\|=\sup\{|\lambda |;\lambda \in \sigma (A)\}}" loading="lazy"></span>.</li></ul>

<p>Es gelten folgende Implikationen:

</p><p>normal <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 \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span> quasinormal <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 \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span> subnormal <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 \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span> hyponormal <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 \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span> paranormal <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 \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span> normaloid.

</p>

<div class="mw-heading mw-heading2"><h2 id="Unbeschränkte_Operatoren"><span id="Unbeschr.C3.A4nkte_Operatoren"></span>Unbeschränkte Operatoren</h2></div>

<p>Ein unbeschränkter Operator <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 A:D(A)\subseteq X\to X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64a68027db22bfa385ad185350c0c3ff950e8a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.829ex; height:2.843ex;" alt="{\displaystyle A:D(A)\subseteq X\to X}" loading="lazy"></span> mit <a href="Definitionsbereich" class="mw-redirect" title="Definitionsbereich">Definitionsbereich</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 D(A)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47f833d059e4565ca5c84985c780b21f1f89f0b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.477ex; height:2.843ex;" alt="{\displaystyle D(A)}" loading="lazy"></span> heißt normal falls

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|Ax\|=\|A^{\ast }x\|,\qquad \forall x\in D(A)=D(A^{\ast })}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e129242fbf945821ad59ad95f0970c0e547f1f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.196ex; height:2.843ex;" alt="{\displaystyle \|Ax\|=\|A^{\ast }x\|,\qquad \forall x\in D(A)=D(A^{\ast })}" loading="lazy"></span></dd></dl>

<p>gilt. Oben genannte äquivalente Charakterisierung der Normalität zeigt, dass es sich um eine Verallgemeinerung der Normalität beschränkter Operatoren handelt. Alle selbstadjungierten Operatoren sind normal, denn für diese gilt <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 A^{\ast }=A}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bd13e94a1712b0dca65d25062c5e94a3039c506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.639ex; height:2.343ex;" alt="{\displaystyle A^{\ast }=A}" loading="lazy"></span>.

</p>

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

<ul><li><a href="Harro_Heuser" title="Harro Heuser">Harro Heuser</a>: <i>Funktionalanalysis</i>. B.G. Teubner, Stuttgart (1986), ISBN 3-519-22206-X.</li>

<li><a href="Gerald_Teschl" title="Gerald Teschl">Gerald Teschl</a>: <i>Mathematical Methods in Quantum Mechanics</i>, American Mathematical Society, Providence (2009), ISBN 978-0-8218-4660-5. (<a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/">freie Online-Version</a>)</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2021-12-24" href="https://de.wikipedia.org/wiki/?title=Normaler_Operator&oldid=218452651">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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