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<title>Normal element</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Normal element</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <a href="Element_(mathematics)" title="Element (mathematics)">element</a> of a <a href="*-algebra" title="*-algebra">*-algebra</a> is called <b>normal</b> if it <a href="Commutative_property" title="Commutative property">commutates</a> with its <span class="nowrap">adjoint.<sup id="cite_ref-FOOTNOTEDixmier19774_1-0" class="reference"><a href="#cite_note-FOOTNOTEDixmier19774-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span>

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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 {\mathcal {A}}}</annotation>

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</math></span><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"></span> be a *-Algebra. An element <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 {A}}}">

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

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</math></span><img src="./6f0b2dd4c933b95137503e49d0f1c12cb8b8b099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.974ex; height:2.343ex;" alt="{\displaystyle a\in {\mathcal {A}}}" loading="lazy"></span> is called normal if it commutes with <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="./5e8ebbdafec4421aba60c540bcd665fe9aaa5c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.343ex;" alt="{\displaystyle a^{*}}" loading="lazy"></span>, i.e. it satisfies the <a href="Equation" title="Equation">equation</a> <span class="nowrap"><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^{*}=a^{*}a}">

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<annotation encoding="application/x-tex">{\displaystyle aa^{*}=a^{*}a}</annotation>

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</math></span><img src="./4b1aee8f7921ab0d61b559b564bb2ece104bb7e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.126ex; height:2.343ex;" alt="{\displaystyle aa^{*}=a^{*}a}" loading="lazy"></span>.</span><sup id="cite_ref-FOOTNOTEDixmier19774_1-1" class="reference"><a href="#cite_note-FOOTNOTEDixmier19774-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of normal elements is denoted by <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 {A}}_{N}}">

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

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</math></span><img src="./9d780278eef5218ca92a2e99ac0ae8bdbd64c199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.546ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{N}}" loading="lazy"></span> or <span class="nowrap"><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 N({\mathcal {A}})}">

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

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</math></span><img src="./2aea68621e48618d00921e2bb12f94b939b936f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.776ex; height:2.843ex;" alt="{\displaystyle N({\mathcal {A}})}" loading="lazy"></span>.</span>

</p><p>A special case of particular importance is the case where <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 {A}}}">

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

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</math></span><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"></span> is a <a href="Banach_algebra#Banach_*-algebras" title="Banach algebra">complete normed *-algebra</a>, that satisfies the C*-identity (<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 \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}</annotation>

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</math></span><img src="./1adbfa98e73900c1abd24c0d1380623e2a4f4b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.393ex; height:3.343ex;" alt="{\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}" loading="lazy"></span>), which is called a <a href="C*-algebra" title="C*-algebra">C*-algebra</a>.

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

<ul><li>Every <a href="Self-adjoint" title="Self-adjoint">self-adjoint element</a> of a a *-algebra is <span class="nowrap">normal.<sup id="cite_ref-FOOTNOTEDixmier19774_1-2" class="reference"><a href="#cite_note-FOOTNOTEDixmier19774-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span></li>

<li>Every <a href="Unitary_element" title="Unitary element">unitary element</a> of a a *-algebra is <span class="nowrap">normal.<sup id="cite_ref-FOOTNOTEDixmier19775_2-0" class="reference"><a href="#cite_note-FOOTNOTEDixmier19775-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span></li>

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

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</math></span><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"></span> is a C*-Algebra and <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 {A}}_{N}}">

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

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</math></span><img src="./9ff660bac54663f606eae99a1461b459f9b319fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.617ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{N}}" loading="lazy"></span> a normal element, then for every <a href="Continuous_function" title="Continuous function">continuous function</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 f}">

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> on the <a href="Banach_algebra#Spectral_theory" title="Banach algebra">spectrum</a> of <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="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> the <a href="Continuous_functional_calculus" title="Continuous functional calculus">continuous functional calculus</a> defines another normal element <span class="nowrap"><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 f(a)}">

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

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</math></span><img src="./368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"></span>.<sup id="cite_ref-FOOTNOTEDixmier197713_3-0" class="reference"><a href="#cite_note-FOOTNOTEDixmier197713-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></span></li></ul>

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

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

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</math></span><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"></span> be a *-algebra. Then:

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<ul><li>An element <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 {A}}}">

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

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</math></span><img src="./6f0b2dd4c933b95137503e49d0f1c12cb8b8b099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.974ex; height:2.343ex;" alt="{\displaystyle a\in {\mathcal {A}}}" loading="lazy"></span> is normal if and only if the *-<a href="Subalgebra" title="Subalgebra">subalgebra</a> generated by <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="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>, meaning the smallest *-algebra containing <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="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>, is <span class="nowrap">commutative.<sup id="cite_ref-FOOTNOTEDixmier19775_2-1" class="reference"><a href="#cite_note-FOOTNOTEDixmier19775-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span></li>

<li>Every element <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 {A}}}">

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

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</math></span><img src="./6f0b2dd4c933b95137503e49d0f1c12cb8b8b099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.974ex; height:2.343ex;" alt="{\displaystyle a\in {\mathcal {A}}}" loading="lazy"></span> can be uniquely decomposed into a <a href="Real_and_imaginary_parts" class="mw-redirect" title="Real and imaginary parts">real and imaginary part</a>, which means there exist self-adjoint elements <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_{1},a_{2}\in {\mathcal {A}}_{sa}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}</annotation>

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</math></span><img src="./12821ffe38f931f9ee8fbd1e96e3ef747169cbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.17ex; height:2.676ex;" alt="{\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}" loading="lazy"></span>, such that <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=a_{1}+\mathrm {i} a_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle a=a_{1}+\mathrm {i} a_{2}}</annotation>

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</math></span><img src="./039ef44511a30468b080076316bb555d31345502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.384ex; height:2.509ex;" alt="{\displaystyle a=a_{1}+\mathrm {i} a_{2}}" loading="lazy"></span>, where <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 \mathrm {i} }">

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

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</math></span><img src="./18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"></span> denotes the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>. Exactly then <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="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> is normal if <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_{1}a_{2}=a_{2}a_{1}}">

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

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</math></span><img src="./7f41f67e1a1e0c4327008796a0abc72c59dd348e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.235ex; height:2.009ex;" alt="{\displaystyle a_{1}a_{2}=a_{2}a_{1}}" loading="lazy"></span>, i.e. real and imaginary part <span class="nowrap">commutate.<sup id="cite_ref-FOOTNOTEDixmier19774_1-3" class="reference"><a href="#cite_note-FOOTNOTEDixmier19774-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span></li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="In_*-algebras">In *-algebras</h3></div>

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

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</math></span><img src="./9ff660bac54663f606eae99a1461b459f9b319fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.617ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{N}}" loading="lazy"></span> be a normal element of a *-algebra <span class="nowrap"><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 {A}}}">

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

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</math></span><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"></span>.</span> Then:

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<ul><li>The adjoint element <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="./5e8ebbdafec4421aba60c540bcd665fe9aaa5c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.343ex;" alt="{\displaystyle a^{*}}" loading="lazy"></span> is also normal, since <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=(a^{*})^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle a=(a^{*})^{*}}</annotation>

</semantics>

</math></span><img src="./dae5268ba667f9282d32e8854ff78d2fb03eb44d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.476ex; height:2.843ex;" alt="{\displaystyle a=(a^{*})^{*}}" loading="lazy"></span> holds for the <a href="Involution_(mathematics)" title="Involution (mathematics)">involution</a> <span class="nowrap">*.<sup id="cite_ref-FOOTNOTEDixmier19773–4_4-0" class="reference"><a href="#cite_note-FOOTNOTEDixmier19773–4-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></span></li></ul>

<div class="mw-heading mw-heading3"><h3 id="In_C*-algebras">In C*-algebras</h3></div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ff660bac54663f606eae99a1461b459f9b319fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.617ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{N}}" loading="lazy"></span> be a normal element of a C*-algebra <span class="nowrap"><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 {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"></span>.</span> Then:

</p>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

<mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\|a^{2}\right\|=\left\|a\right\|^{2}}</annotation>

</semantics>

</math></span><img src="./b8ff1f1846ea59da48f0d6ac3385a12e157736d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.316ex; height:3.676ex;" alt="{\displaystyle \left\|a^{2}\right\|=\left\|a\right\|^{2}}" loading="lazy"></span>, since for normal elements using the C*-identity <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 \left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}}">

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<msup>

<mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}}</annotation>

</semantics>

</math></span><img src="./1c9a93582ba42dd58b3c963ba712500b46df04b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.98ex; height:5.176ex;" alt="{\displaystyle \left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}}" loading="lazy"></span> <span class="nowrap">holds.<sup id="cite_ref-FOOTNOTEWerner2018518_5-0" class="reference"><a href="#cite_note-FOOTNOTEWerner2018518-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></span></li>

<li>Every normal element is a normaloid element, i.e. the <a href="Spectral_radius" title="Spectral radius">spectral radius</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 r(a)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./43025a21c70d2da843f3a0e0e99bb6a8396d1963.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.088ex; height:2.843ex;" alt="{\displaystyle r(a)}" loading="lazy"></span> equals the norm of <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>, i.e. <span class="nowrap"><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 r(a)=\left\|a\right\|}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r(a)=\left\|a\right\|}</annotation>

</semantics>

</math></span><img src="./ed55df137815cc9453c4b4c6d36227d01e3749af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle r(a)=\left\|a\right\|}" loading="lazy"></span>.<sup id="cite_ref-FOOTNOTEHeuser1982390_6-0" class="reference"><a href="#cite_note-FOOTNOTEHeuser1982390-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></span> This follows from the <a href="Spectral_radius#Gelfand's_formula" title="Spectral radius">spectral radius formula</a> by repeated application of the previous property.<sup id="cite_ref-FOOTNOTEWerner2018284–285,_518_7-0" class="reference"><a href="#cite_note-FOOTNOTEWerner2018284–285,_518-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> to <span class="nowrap"><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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>.<sup id="cite_ref-FOOTNOTEDixmier197713_3-1" class="reference"><a href="#cite_note-FOOTNOTEDixmier197713-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></span></li></ul>

<ul><li><a href="Normal_matrix" title="Normal matrix">Normal matrix</a></li>

<li><a href="Normal_operator" title="Normal operator">Normal operator</a></li></ul>

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

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</style><div class="reflist">

<li id="cite_note-FOOTNOTEDixmier19774-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDixmier19774_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDixmier19774_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDixmier19774_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDixmier19774_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, p. 4.</span>

</li>

<li id="cite_note-FOOTNOTEDixmier19775-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDixmier19775_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDixmier19775_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, p. 5.</span>

</li>

<li id="cite_note-FOOTNOTEDixmier197713-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDixmier197713_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDixmier197713_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, p. 13.</span>

</li>

<li id="cite_note-FOOTNOTEDixmier19773–4-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDixmier19773–4_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, pp. 3–4.</span>

</li>

<li id="cite_note-FOOTNOTEWerner2018518-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWerner2018518_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWerner2018">Werner 2018</a>, p. 518.</span>

</li>

<li id="cite_note-FOOTNOTEHeuser1982390-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHeuser1982390_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeuser1982">Heuser 1982</a>, p. 390.</span>

</li>

<li id="cite_note-FOOTNOTEWerner2018284–285,_518-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWerner2018284–285,_518_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWerner2018">Werner 2018</a>, pp. 284–285, 518.</span>

</li>

</ol></div></div>

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

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFDixmier1977" class="citation book cs1">Dixmier, Jacques (1977). <i>C*-algebras</i>. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7204-0762-1</bdi>.</cite> English translation of <cite id="CITEREFDixmier1969" class="citation book cs1 cs1-prop-foreign-lang-source"><i>Les C*-algèbres et leurs représentations</i> (in French). Gauthier-Villars. 1969.</cite></li>

<li><cite id="CITEREFHeuser1982" class="citation book cs1">Heuser, Harro (1982). <i>Functional analysis</i>. Translated by Horvath, John. John Wiley & Sons Ltd. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-10069-2</bdi>.</cite></li>

<li><cite id="CITEREFWerner2018" class="citation book cs1 cs1-prop-foreign-lang-source">Werner, Dirk (2018). <i>Funktionalanalysis</i> (in German) (8 ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-55407-4</bdi>.</cite></li></ul>

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</style><div id="Spectral_theory_and_*-algebras154" style="font-size:114%;margin:0 4em"><a href="Spectral_theory" title="Spectral theory">Spectral theory</a> and <a href="*-algebra" title="*-algebra"><sup>*</sup>-algebras</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="*-algebra" title="*-algebra">Involution/*-algebra</a></li>

<li><a href="Banach_algebra" title="Banach algebra">Banach algebra</a></li>

<li><a href="Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">B*-algebra</a></li>

<li><a href="C*-algebra" title="C*-algebra">C*-algebra</a></li>

<li><a href="Noncommutative_topology" title="Noncommutative topology">Noncommutative topology</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li>

<li><a href="Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></li>

<li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li>

<li><a href="Spectral_radius" title="Spectral radius">Spectral radius</a></li>

<li><a href="Operator_space" title="Operator space">Operator space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur theorem</a></li>

<li><a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a></li>

<li><a href="Gelfand_representation" title="Gelfand representation">Gelfand representation</a></li>

<li><a href="Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li>

<li><a href="Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li>

<li><a href="Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li>

<li><a href="Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special Elements/Operators</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Isospectral" title="Isospectral">Isospectral</a></li>

<li> <a href="Normal_operator" title="Normal operator">operator</a></li>

<li><a href="Self-adjoint" title="Self-adjoint">Hermitian/Self-adjoint</a> <a href="Self-adjoint_operator" title="Self-adjoint operator">operator</a></li>

<li><a href="Unitary_element" title="Unitary element">Unitary</a> <a href="Unitary_operator" title="Unitary operator">operator</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Krein%E2%80%93Rutman_theorem" title="Krein–Rutman theorem">Krein–Rutman theorem</a></li>

<li><a href="Normal_eigenvalue" title="Normal eigenvalue">Normal eigenvalue</a></li>

<li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li>

<li><a href="Spectral_radius" title="Spectral radius">Spectral radius</a></li>

<li><a href="Spectral_asymmetry" title="Spectral asymmetry">Spectral asymmetry</a></li>

<li><a href="Spectral_gap" title="Spectral gap">Spectral gap</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decomposition</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_spectrum_(functional_analysis)" title="Decomposition of spectrum (functional analysis)">Decomposition of a spectrum</a>

<ul><li><a href="Continuous_spectrum_(functional_analysis)" class="mw-redirect" title="Continuous spectrum (functional analysis)">Continuous</a></li>

<li><a href="Point_spectrum" class="mw-redirect" title="Point spectrum">Point</a></li>

<li><a href="Spectrum_(functional_analysis)#Residual_spectrum" title="Spectrum (functional analysis)">Residual</a></li></ul></li>

<li><a href="Spectrum_(functional_analysis)#Approximate_point_spectrum" title="Spectrum (functional analysis)">Approximate point</a></li>

<li><a href="Spectrum_(functional_analysis)#Compression_spectrum" title="Spectrum (functional analysis)">Compression</a></li>

<li><a href="Direct_integral" title="Direct integral">Direct integral</a></li>

<li><a href="Discrete_spectrum_(mathematics)" title="Discrete spectrum (mathematics)">Discrete</a></li>

<li><a href="Spectral_abscissa" title="Spectral abscissa">Spectral abscissa</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spectral Theorem</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li>

<li><a href="Min-max_theorem" title="Min-max theorem">Min-max theorem</a></li>

<li><a href="Positive_operator-valued_measure" class="mw-redirect" title="Positive operator-valued measure">Positive operator-valued measure</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li>

<li><a href="Riesz_projector" title="Riesz projector">Riesz projector</a></li>

<li><a href="Rigged_Hilbert_space" title="Rigged Hilbert space">Rigged Hilbert space</a></li>

<li><a href="Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li>

<li><a href="Spectral_theory_of_compact_operators" title="Spectral theory of compact operators">Spectral theory of compact operators</a></li>

<li><a href="Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amenable_Banach_algebra" title="Amenable Banach algebra">Amenable Banach algebra</a></li>

<li>With an <a href="Approximate_identity" title="Approximate identity">Approximate identity</a></li>

<li><a href="Banach_function_algebra" title="Banach function algebra">Banach function algebra</a></li>

<li><a href="Disk_algebra" title="Disk algebra">Disk algebra</a></li>

<li><a href="Nuclear_C*-algebra" title="Nuclear C*-algebra">Nuclear C*-algebra</a></li>

<li><a href="Uniform_algebra" title="Uniform algebra">Uniform algebra</a></li>

<li><a href="Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a>

<ul><li><a href="Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Finite-Dimensional</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alon%E2%80%93Boppana_bound" title="Alon–Boppana bound">Alon–Boppana bound</a></li>

<li><a href="Bauer%E2%80%93Fike_theorem" title="Bauer–Fike theorem">Bauer–Fike theorem</a></li>

<li><a href="Numerical_range" title="Numerical range">Numerical range</a></li>

<li><a href="Schur%E2%80%93Horn_theorem" title="Schur–Horn theorem">Schur–Horn theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirac_spectrum" title="Dirac spectrum">Dirac spectrum</a></li>

<li><a href="Essential_spectrum" title="Essential spectrum">Essential spectrum</a></li>

<li><a href="Pseudospectrum" title="Pseudospectrum">Pseudospectrum</a></li>

<li><a href="Structure_space" class="mw-redirect" title="Structure space">Structure space</a> (<a href="Shilov_boundary" title="Shilov boundary">Shilov boundary</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_index_group" class="mw-redirect" title="Abstract index group">Abstract index group</a></li>

<li><a href="Banach_algebra_cohomology" title="Banach algebra cohomology">Banach algebra cohomology</a></li>

<li><a href="Cohen%E2%80%93Hewitt_factorization_theorem" title="Cohen–Hewitt factorization theorem">Cohen–Hewitt factorization theorem</a></li>

<li><a href="Extensions_of_symmetric_operators" title="Extensions of symmetric operators">Extensions of symmetric operators</a></li>

<li><a href="Fredholm_theory" title="Fredholm theory">Fredholm theory</a></li>

<li><a href="Limiting_absorption_principle" title="Limiting absorption principle">Limiting absorption principle</a></li>

<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorems_for_operator_algebras" title="Schröder–Bernstein theorems for operator algebras">Schröder–Bernstein theorems for operator algebras</a></li>

<li><a href="Sherman%E2%80%93Takeda_theorem" title="Sherman–Takeda theorem">Sherman–Takeda theorem</a></li>

<li><a href="Unbounded_operator" title="Unbounded operator">Unbounded operator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wiener_algebra" title="Wiener algebra">Wiener algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_Mathieu_operator" title="Almost Mathieu operator">Almost Mathieu operator</a></li>

<li><a href="Corona_theorem" title="Corona theorem">Corona theorem</a></li>

<li><a href="Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">Hearing the shape of a drum</a> (<a href="Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalue</a>)</li>

<li><a href="Heat_kernel" title="Heat kernel">Heat kernel</a></li>

<li><a href="Kuznetsov_trace_formula" title="Kuznetsov trace formula">Kuznetsov trace formula</a></li>

<li><a href="Proto-value_function" title="Proto-value function">Proto-value function</a></li>

<li><a href="Ramanujan_graph" title="Ramanujan graph">Ramanujan graph</a></li>

<li><a href="Rayleigh%E2%80%93Faber%E2%80%93Krahn_inequality" title="Rayleigh–Faber–Krahn inequality">Rayleigh–Faber–Krahn inequality</a></li>

<li><a href="Spectral_geometry" title="Spectral geometry">Spectral geometry</a></li>

<li><a href="Spectral_method" title="Spectral method">Spectral method</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li>

<li><a href="Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li>

<li><a href="Superstrong_approximation" title="Superstrong approximation">Superstrong approximation</a></li>

<li><a href="Transfer_operator" title="Transfer operator">Transfer operator</a></li>

<li><a href="Transform_theory" title="Transform theory">Transform theory</a></li>

<li><a href="Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-03-10" href="https://en.wikipedia.org/wiki/?title=Normal_element&oldid=1212959667">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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