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<b>self-adjoint</b> if it is the same as its adjoint (i.e. <span class="mwe-math-element"><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^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a41a09988020998866e69b7212eb9e72e6aee263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.612ex; height:2.343ex;" alt="{\displaystyle a=a^{*}}"></span>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> be a *-algebra. An element <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0b2dd4c933b95137503e49d0f1c12cb8b8b099" 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}}}"></span> is called self-adjoint if <span class="nowrap"><span class="mwe-math-element"><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^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a41a09988020998866e69b7212eb9e72e6aee263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.612ex; height:2.343ex;" alt="{\displaystyle a=a^{*}}"></span>.<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> </p><p>The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of self-adjoint elements is referred to as <span class="nowrap"><span class="mwe-math-element"><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}}_{sa}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b5927234665ebef064b45c7316d4d029d89f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.728ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{sa}}"></span>.</span> </p><p>A <a href="/wiki/Subset" title="Subset">subset</a> <span class="mwe-math-element"><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 {B}}\subseteq {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad0aac444d2da7ff64650540247d945b0fc3524c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}}"></span> that is <a href="/wiki/Closed_set" title="Closed set">closed</a> under the <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> *, i.e. <span class="mwe-math-element"><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 {B}}={\mathcal {B}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f9d0f4df7129d12ce27a75e91daa44e8a46916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.244ex; height:2.343ex;" alt="{\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}}"></span>, is called <span class="nowrap">self-adjoint.<sup id="cite_ref-FOOTNOTEDixmier19773_2-0" class="reference"><a href="#cite_note-FOOTNOTEDixmier19773-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span> </p><p>A special case of particular importance is the case where <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> is a <a href="/wiki/Banach_algebra#Banach_*-algebras" title="Banach algebra">complete normed *-algebra</a>, that satisfies the C*-identity (<span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mi>a</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adbfa98e73900c1abd24c0d1380623e2a4f4b33" 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}}}"></span>), which is called a <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a>. </p><p>Especially in the older literature on *-algebras and C*-algebras, such elements are often called <span class="nowrap">hermitian.<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></span> Because of that the notations <span class="mwe-math-element"><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}}_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f4a33a26a9d269e1a664265abe54dc6c2d4955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.034ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{h}}"></span>, <span class="mwe-math-element"><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}}_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2a8e16ba0893bcee28a0a71d586b40b9ae2d68" 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}}_{H}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H({\mathcal {A}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H({\mathcal {A}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82aa0218143db8fe29094758a52b00b3aea3f58" 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 H({\mathcal {A}})}"></span> for the set of self-adjoint elements are also sometimes used, even in the more recent literature. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Each <a href="/wiki/Positive_element" title="Positive element">positive element</a> of a C*-algebra is <span class="nowrap">self-adjoint.<sup id="cite_ref-FOOTNOTEPalmer2001800_3-0" class="reference"><a href="#cite_note-FOOTNOTEPalmer2001800-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></span></li> <li>For each element <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></span> of a *-algebra, the elements <span class="mwe-math-element"><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^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aa^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3c4b0cad7eb46eb0874748f1d123b4902a9e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.514ex; height:2.343ex;" alt="{\displaystyle aa^{*}}"></span> and <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7577c4399031e6258509f27b076498aaab0f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.514ex; height:2.343ex;" alt="{\displaystyle a^{*}a}"></span> are self-adjoint, since * is an <span class="nowrap"><a href="/wiki/Antihomomorphism#Involutions" title="Antihomomorphism">involutive antiautomorphism</a>.<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> <li>For each element <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></span> of a *-algebra, the <a href="/wiki/Real_and_imaginary_parts" class="mw-redirect" title="Real and imaginary parts">real and imaginary parts</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4595543e41e71f5dbe827ca42591e1b8440e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.702ex; height:3.509ex;" alt="{\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9bdec633cb1243c63ddced005cc80dd9db2c686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.192ex; height:3.509ex;" alt="{\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})}"></span> are self-adjoint, where <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f0f09f6fc40e634d34aed6e205ac0f7a40e062" 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} }"></span> denotes the <span class="nowrap"><a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>.<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>If <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff660bac54663f606eae99a1461b459f9b319fc" 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}}"></span> is a <a href="/wiki/Normal_element" title="Normal element">normal element</a> of a C*-algebra <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span>, then for every <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"></span>, which is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> on the <a href="/wiki/Banach_algebra#Spectral_theory" title="Banach algebra">spectrum</a> of <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></span>, the <a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous functional calculus</a> defines a self-adjoint element <span class="nowrap"><span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368cb4b81ba5754d7a354a4ce49c2f1084bdaace" 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)}"></span>.<sup id="cite_ref-FOOTNOTEKadisonRingrose1983271_5-0" class="reference"><a href="#cite_note-FOOTNOTEKadisonRingrose1983271-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Criteria">Criteria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=3" title="Edit section: Criteria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> be a *-algebra. Then: </p> <ul><li>Let <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0b2dd4c933b95137503e49d0f1c12cb8b8b099" 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}}}"></span>, then <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7577c4399031e6258509f27b076498aaab0f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.514ex; height:2.343ex;" alt="{\displaystyle a^{*}a}"></span> is self-adjoint, since <span class="mwe-math-element"><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)^{*}=a^{*}(a^{*})^{*}=a^{*}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d7932cf81990d9ad4bcd7f0cd24f5520666b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.52ex; height:2.843ex;" alt="{\displaystyle (a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a}"></span>. A similarly calculation yields that <span class="mwe-math-element"><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^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aa^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3c4b0cad7eb46eb0874748f1d123b4902a9e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.514ex; height:2.343ex;" alt="{\displaystyle aa^{*}}"></span> is also <span class="nowrap">self-adjoint.<sup id="cite_ref-FOOTNOTEPalmer2001798–800_6-0" class="reference"><a href="#cite_note-FOOTNOTEPalmer2001798–800-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></span></li> <li>Let <span class="mwe-math-element"><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}a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a_{1}a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e5649295a2a6950e27c461d5f20f297f7ac4df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.896ex; height:2.009ex;" alt="{\displaystyle a=a_{1}a_{2}}"></span> be the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of two self-adjoint elements <span class="nowrap"><span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12821ffe38f931f9ee8fbd1e96e3ef747169cbfa" 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}}"></span>.</span> Then <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></span> is self-adjoint if <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></span> and <span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}"></span> <a href="/wiki/Commutative_property" title="Commutative property">commutate</a>, since <span class="mwe-math-element"><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}^{*}=a_{2}a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}a_{2})^{*}=a_{2}^{*}a_{1}^{*}=a_{2}a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87c878fc313acbb6fa7744be8ca6173b9de957d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.765ex; height:3.009ex;" alt="{\displaystyle (a_{1}a_{2})^{*}=a_{2}^{*}a_{1}^{*}=a_{2}a_{1}}"></span> always <span class="nowrap">holds.<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> <li>If <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> is a C*-algebra, then a normal element <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff660bac54663f606eae99a1461b459f9b319fc" 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}}"></span> is self-adjoint if and only if its spectrum is real, i.e. <span class="nowrap"><span class="mwe-math-element"><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)\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (a)\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7384d04f41477f3beb31b068a3fad0e781b496f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.843ex;" alt="{\displaystyle \sigma (a)\subseteq \mathbb {R} }"></span>.<sup id="cite_ref-FOOTNOTEKadisonRingrose1983271_5-1" class="reference"><a href="#cite_note-FOOTNOTEKadisonRingrose1983271-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_*-algebras"><span id="In_.2A-algebras"></span>In *-algebras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=5" title="Edit section: In *-algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> be a *-algebra. Then: </p> <ul><li>Each element <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0b2dd4c933b95137503e49d0f1c12cb8b8b099" 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}}}"></span> can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12821ffe38f931f9ee8fbd1e96e3ef747169cbfa" 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}}"></span>, so that <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a_{1}+\mathrm {i} a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/039ef44511a30468b080076316bb555d31345502" 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}}"></span> holds. Where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{1}={\frac {1}{2}}(a+a^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a_{1}={\frac {1}{2}}(a+a^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cbc64e4e1a67e942e7ccd6abcbc29c845045f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.204ex; height:3.509ex;" alt="{\textstyle a_{1}={\frac {1}{2}}(a+a^{*})}"></span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3f4423cffde132081173c95e8f7c9dcd454b56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.662ex; height:3.509ex;" alt="{\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})}"></span>.<sup id="cite_ref-FOOTNOTEDixmier19774_1-4" class="reference"><a href="#cite_note-FOOTNOTEDixmier19774-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span></li> <li>The set of self-adjoint elements <span class="mwe-math-element"><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}}_{sa}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b5927234665ebef064b45c7316d4d029d89f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.728ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{sa}}"></span> is a <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <span class="nowrap"><span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span>.</span> From the previous property, it follows that <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> is the <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a> of two real linear subspaces, i.e. <span class="nowrap"><span class="mwe-math-element"><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}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65fdbbc019dbe9401b549114499c03a3251da7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.944ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}}"></span>.<sup id="cite_ref-FOOTNOTEPalmer2001798_7-0" class="reference"><a href="#cite_note-FOOTNOTEPalmer2001798-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></span></li> <li>If <span class="mwe-math-element"><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}}_{sa}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b4db3cc20b971f2c1e537eac5dcd563065f82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.798ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{sa}}"></span> is self-adjoint, then <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></span> is <span class="nowrap">normal.<sup id="cite_ref-FOOTNOTEDixmier19774_1-5" class="reference"><a href="#cite_note-FOOTNOTEDixmier19774-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span></li> <li>The *-algebra <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> is called a hermitian *-algebra if every self-adjoint element <span class="mwe-math-element"><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}}_{sa}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b4db3cc20b971f2c1e537eac5dcd563065f82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.798ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{sa}}"></span> has a real spectrum <span class="nowrap"><span class="mwe-math-element"><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)\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (a)\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7384d04f41477f3beb31b068a3fad0e781b496f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.843ex;" alt="{\displaystyle \sigma (a)\subseteq \mathbb {R} }"></span>.<sup id="cite_ref-FOOTNOTEPalmer20011008_8-0" class="reference"><a href="#cite_note-FOOTNOTEPalmer20011008-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="In_C*-algebras"><span id="In_C.2A-algebras"></span>In C*-algebras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=6" title="Edit section: In C*-algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"></span> be a C*-algebra and <span class="mwe-math-element"><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}}_{sa}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}_{sa}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b4db3cc20b971f2c1e537eac5dcd563065f82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.798ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{sa}}"></span>. Then: </p> <ul><li>For the spectrum <span class="mwe-math-element"><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\right\|\in \sigma (a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mi>a</mi> <mo symmetric="true">‖</mo> </mrow> <mo>∈</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|a\right\|\in \sigma (a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80def6cd7920b4adaf68fa1cd03c27f053454ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.764ex; height:2.843ex;" alt="{\displaystyle \left\|a\right\|\in \sigma (a)}"></span> or <span class="mwe-math-element"><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\right\|\in \sigma (a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow> <mo symmetric="true">‖</mo> <mi>a</mi> <mo symmetric="true">‖</mo> </mrow> <mo>∈</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\left\|a\right\|\in \sigma (a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5999637a526bdcd79185a3103bdc8c22dda085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.959ex; height:2.843ex;" alt="{\displaystyle -\left\|a\right\|\in \sigma (a)}"></span> holds, since <span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9ea3f420f9f4df8439a86b1256e4157c632e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.369ex; height:2.843ex;" alt="{\displaystyle \sigma (a)}"></span> is real and <span class="mwe-math-element"><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\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mi>a</mi> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(a)=\left\|a\right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed55df137815cc9453c4b4c6d36227d01e3749af" 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\|}"></span> holds for the <a href="/wiki/Banach_algebra#Spectral_theory" title="Banach algebra">spectral radius</a>, because <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></span> is <span class="nowrap">normal.<sup id="cite_ref-FOOTNOTEKadisonRingrose1983238_9-0" class="reference"><a href="#cite_note-FOOTNOTEKadisonRingrose1983238-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></span></li> <li>According to the continuous functional calculus, there exist uniquely determined positive elements <span class="mwe-math-element"><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_{-}\in {\mathcal {A}}_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{+},a_{-}\in {\mathcal {A}}_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c98395177d764570099e447385265e2f5b90b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.721ex; height:2.676ex;" alt="{\displaystyle a_{+},a_{-}\in {\mathcal {A}}_{+}}"></span>, such that <span class="mwe-math-element"><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_{+}-a_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a_{+}-a_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48360a6c23d1f2209dd567218367799b225adf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.65ex; height:2.343ex;" alt="{\displaystyle a=a_{+}-a_{-}}"></span> with <span class="nowrap"><span class="mwe-math-element"><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_{-}=a_{-}a_{+}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{+}a_{-}=a_{-}a_{+}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0660a092799aebb9115f9c4748bb3a8d882bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.322ex; height:2.509ex;" alt="{\displaystyle a_{+}a_{-}=a_{-}a_{+}=0}"></span>.</span> For the norm, <span class="mwe-math-element"><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\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mi>a</mi> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mo>,</mo> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494440b56f1d15045bed097d546644c3b4d334ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.34ex; height:2.843ex;" alt="{\displaystyle \left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)}"></span> holds.<sup id="cite_ref-FOOTNOTEKadisonRingrose1983246_10-0" class="reference"><a href="#cite_note-FOOTNOTEKadisonRingrose1983246-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The elements <span class="mwe-math-element"><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_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4cdfdba95dad3529aa2e0783642a989c6c9624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.741ex; height:2.009ex;" alt="{\displaystyle a_{+}}"></span> and <span class="mwe-math-element"><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_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631ec2342e069d496e3a96a759ee8bccbe45bf33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.741ex; height:2.009ex;" alt="{\displaystyle a_{-}}"></span> are also referred to as the <a href="/wiki/Positive_and_negative_parts" title="Positive and negative parts">positive and negative parts</a>. In addition, <span class="mwe-math-element"><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_{+}+a_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|=a_{+}+a_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9387ce03aa8d11632e316c400f3b2fb8555e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.944ex; height:2.843ex;" alt="{\displaystyle |a|=a_{+}+a_{-}}"></span> holds for the absolute value defined for every element <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5161a47297f2f93d6660dfb56d75326e360d1727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.681ex; height:4.009ex;" alt="{\textstyle |a|=(a^{*}a)^{\frac {1}{2}}}"></span>.<sup id="cite_ref-FOOTNOTEDixmier197715_11-0" class="reference"><a href="#cite_note-FOOTNOTEDixmier197715-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></span></li> <li>For every <span class="mwe-math-element"><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}}_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {A}}_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33ae19320ca4f3c182d74c9da9671fa7b86ce5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.436ex; height:2.676ex;" alt="{\displaystyle a\in {\mathcal {A}}_{+}}"></span> and odd <span class="mwe-math-element"><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\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>, there exists a uniquely determined <span class="mwe-math-element"><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 {A}}_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in {\mathcal {A}}_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ecdb434d4b29cc37b96d31b3b6bc1edc616b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.204ex; height:2.676ex;" alt="{\displaystyle b\in {\mathcal {A}}_{+}}"></span> that satisfies <span class="mwe-math-element"><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^{n}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac985eb2c832100255a52f9f170e5c34c1b08fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.544ex; height:2.343ex;" alt="{\displaystyle b^{n}=a}"></span>, i.e. a unique <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><a href="/wiki/Nth_root" title="Nth root">-th root</a>, as can be shown with the continuous functional <span class="nowrap">calculus.<sup id="cite_ref-FOOTNOTEBlackadar200663_12-0" class="reference"><a href="#cite_note-FOOTNOTEBlackadar200663-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Self-adjoint matrix</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint operator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <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> <a href="#cite_ref-FOOTNOTEDixmier19774_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDixmier19774_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, p. 4.</span> </li> <li id="cite_note-FOOTNOTEDixmier19773-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDixmier19773_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, p. 3.</span> </li> <li id="cite_note-FOOTNOTEPalmer2001800-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPalmer2001800_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPalmer2001">Palmer 2001</a>, p. 800.</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-FOOTNOTEKadisonRingrose1983271-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEKadisonRingrose1983271_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKadisonRingrose1983271_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKadisonRingrose1983">Kadison & Ringrose 1983</a>, p. 271.</span> </li> <li id="cite_note-FOOTNOTEPalmer2001798–800-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPalmer2001798–800_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPalmer2001">Palmer 2001</a>, pp. 798–800.</span> </li> <li id="cite_note-FOOTNOTEPalmer2001798-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPalmer2001798_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPalmer2001">Palmer 2001</a>, p. 798.</span> </li> <li id="cite_note-FOOTNOTEPalmer20011008-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPalmer20011008_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPalmer2001">Palmer 2001</a>, p. 1008.</span> </li> <li id="cite_note-FOOTNOTEKadisonRingrose1983238-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKadisonRingrose1983238_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKadisonRingrose1983">Kadison & Ringrose 1983</a>, p. 238.</span> </li> <li id="cite_note-FOOTNOTEKadisonRingrose1983246-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKadisonRingrose1983246_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKadisonRingrose1983">Kadison & Ringrose 1983</a>, p. 246.</span> </li> <li id="cite_note-FOOTNOTEDixmier197715-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDixmier197715_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDixmier1977">Dixmier 1977</a>, p. 15.</span> </li> <li id="cite_note-FOOTNOTEBlackadar200663-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlackadar200663_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlackadar2006">Blackadar 2006</a>, p. 63.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-adjoint&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBlackadar2006" class="citation book cs1">Blackadar, Bruce (2006). <i>Operator Algebras. Theory of C*-Algebras and von Neumann Algebras</i>. Berlin/Heidelberg: Springer. p. 63. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-28486-9" title="Special:BookSources/3-540-28486-9"><bdi>3-540-28486-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operator+Algebras.+Theory+of+C%2A-Algebras+and+von+Neumann+Algebras&rft.place=Berlin%2FHeidelberg&rft.pages=63&rft.pub=Springer&rft.date=2006&rft.isbn=3-540-28486-9&rft.aulast=Blackadar&rft.aufirst=Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASelf-adjoint" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7204-0762-1" title="Special:BookSources/0-7204-0762-1"><bdi>0-7204-0762-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C%2A-algebras&rft.place=Amsterdam%2FNew+York%2FOxford&rft.pub=North-Holland&rft.date=1977&rft.isbn=0-7204-0762-1&rft.aulast=Dixmier&rft.aufirst=Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASelf-adjoint" class="Z3988"></span> English translation of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Les+C%2A-alg%C3%A8bres+et+leurs+repr%C3%A9sentations&rft.pub=Gauthier-Villars&rft.date=1969&rft.aulast=Dixmier&rft.aufirst=Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASelf-adjoint" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKadisonRingrose1983" class="citation book cs1">Kadison, Richard V.; Ringrose, John R. (1983). <i>Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory</i>. New York/London: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-393301-3" title="Special:BookSources/0-12-393301-3"><bdi>0-12-393301-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+the+Theory+of+Operator+Algebras.+Volume+1+Elementary+Theory.&rft.place=New+York%2FLondon&rft.pub=Academic+Press&rft.date=1983&rft.isbn=0-12-393301-3&rft.aulast=Kadison&rft.aufirst=Richard+V.&rft.au=Ringrose%2C+John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASelf-adjoint" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPalmer2001" class="citation book cs1">Palmer, Theodore W. (2001). <i>Banach algebras and the general theory of*-algebras: Volume 2,*-algebras</i>. Cambridge university press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-36638-0" title="Special:BookSources/0-521-36638-0"><bdi>0-521-36638-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Banach+algebras+and+the+general+theory+of%2A-algebras%3A+Volume+2%2C%2A-algebras.&rft.pub=Cambridge+university+press&rft.date=2001&rft.isbn=0-521-36638-0&rft.aulast=Palmer&rft.aufirst=Theodore+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASelf-adjoint" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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href="/wiki/Template:Spectral_theory" title="Template:Spectral theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Spectral_theory" title="Template talk:Spectral theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Spectral_theory" title="Special:EditPage/Template:Spectral theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Spectral_theory_and_*-algebras" style="font-size:114%;margin:0 4em"><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> and <a href="/wiki/*-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="/wiki/*-algebra" title="*-algebra">Involution/*-algebra</a></li> <li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">B*-algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Noncommutative_topology" title="Noncommutative topology">Noncommutative topology</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/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="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur theorem</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a></li> <li><a href="/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/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="/wiki/Isospectral" title="Isospectral">Isospectral</a></li> <li><a href="/wiki/Normal_element" title="Normal element">Normal</a> <a href="/wiki/Normal_operator" title="Normal operator">operator</a></li> <li><a class="mw-selflink selflink">Hermitian/Self-adjoint</a> <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">operator</a></li> <li><a href="/wiki/Unitary_element" title="Unitary element">Unitary</a> <a href="/wiki/Unitary_operator" title="Unitary operator">operator</a></li> <li><a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">Unit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Krein%E2%80%93Rutman_theorem" title="Krein–Rutman theorem">Krein–Rutman theorem</a></li> <li><a href="/wiki/Normal_eigenvalue" title="Normal eigenvalue">Normal eigenvalue</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/Spectral_asymmetry" title="Spectral asymmetry">Spectral asymmetry</a></li> <li><a href="/wiki/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="/wiki/Decomposition_of_spectrum_(functional_analysis)" title="Decomposition of spectrum (functional analysis)">Decomposition of a spectrum</a> <ul><li><a href="/wiki/Continuous_spectrum_(functional_analysis)" class="mw-redirect" title="Continuous spectrum (functional analysis)">Continuous</a></li> <li><a href="/wiki/Point_spectrum" class="mw-redirect" title="Point spectrum">Point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Residual_spectrum" title="Spectrum (functional analysis)">Residual</a></li></ul></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Approximate_point_spectrum" title="Spectrum (functional analysis)">Approximate point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Compression_spectrum" title="Spectrum (functional analysis)">Compression</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Discrete_spectrum_(mathematics)" title="Discrete spectrum (mathematics)">Discrete</a></li> <li><a href="/wiki/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="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min-max theorem</a></li> <li><a href="/wiki/Positive_operator-valued_measure" class="mw-redirect" title="Positive operator-valued measure">Positive operator-valued measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Riesz_projector" title="Riesz projector">Riesz projector</a></li> <li><a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">Rigged Hilbert space</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_compact_operators" title="Spectral theory of compact operators">Spectral theory of compact operators</a></li> <li><a href="/wiki/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="/wiki/Amenable_Banach_algebra" title="Amenable Banach algebra">Amenable Banach algebra</a></li> <li>With an <a href="/wiki/Approximate_identity" title="Approximate identity">Approximate identity</a></li> <li><a href="/wiki/Banach_function_algebra" title="Banach function algebra">Banach function algebra</a></li> <li><a href="/wiki/Disk_algebra" title="Disk algebra">Disk algebra</a></li> <li><a href="/wiki/Nuclear_C*-algebra" title="Nuclear C*-algebra">Nuclear C*-algebra</a></li> <li><a href="/wiki/Uniform_algebra" title="Uniform algebra">Uniform algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a> <ul><li><a href="/wiki/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="/wiki/Alon%E2%80%93Boppana_bound" title="Alon–Boppana bound">Alon–Boppana bound</a></li> <li><a href="/wiki/Bauer%E2%80%93Fike_theorem" title="Bauer–Fike theorem">Bauer–Fike theorem</a></li> <li><a href="/wiki/Numerical_range" title="Numerical range">Numerical range</a></li> <li><a href="/wiki/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="/wiki/Dirac_spectrum" title="Dirac spectrum">Dirac spectrum</a></li> <li><a href="/wiki/Essential_spectrum" title="Essential spectrum">Essential spectrum</a></li> <li><a href="/wiki/Pseudospectrum" title="Pseudospectrum">Pseudospectrum</a></li> <li><a href="/wiki/Structure_space" class="mw-redirect" title="Structure space">Structure space</a> (<a href="/wiki/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="/wiki/Abstract_index_group" class="mw-redirect" title="Abstract index group">Abstract index group</a></li> <li><a href="/wiki/Banach_algebra_cohomology" title="Banach algebra cohomology">Banach algebra cohomology</a></li> <li><a href="/wiki/Cohen%E2%80%93Hewitt_factorization_theorem" title="Cohen–Hewitt factorization theorem">Cohen–Hewitt factorization theorem</a></li> <li><a href="/wiki/Extensions_of_symmetric_operators" title="Extensions of symmetric operators">Extensions of symmetric operators</a></li> <li><a href="/wiki/Fredholm_theory" title="Fredholm theory">Fredholm theory</a></li> <li><a href="/wiki/Limiting_absorption_principle" title="Limiting absorption principle">Limiting absorption principle</a></li> <li><a href="/wiki/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="/wiki/Sherman%E2%80%93Takeda_theorem" title="Sherman–Takeda theorem">Sherman–Takeda theorem</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Almost_Mathieu_operator" title="Almost Mathieu operator">Almost Mathieu operator</a></li> <li><a href="/wiki/Corona_theorem" title="Corona theorem">Corona theorem</a></li> <li><a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">Hearing the shape of a drum</a> (<a href="/wiki/Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalue</a>)</li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Kuznetsov_trace_formula" title="Kuznetsov trace formula">Kuznetsov trace formula</a></li> <li><a href="/wiki/Lax_pair" title="Lax pair">Lax pair</a></li> <li><a href="/wiki/Proto-value_function" title="Proto-value function">Proto-value function</a></li> <li><a href="/wiki/Ramanujan_graph" title="Ramanujan graph">Ramanujan graph</a></li> <li><a href="/wiki/Rayleigh%E2%80%93Faber%E2%80%93Krahn_inequality" title="Rayleigh–Faber–Krahn inequality">Rayleigh–Faber–Krahn inequality</a></li> <li><a href="/wiki/Spectral_geometry" title="Spectral geometry">Spectral geometry</a></li> <li><a href="/wiki/Spectral_method" title="Spectral method">Spectral method</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li> <li><a href="/wiki/Superstrong_approximation" title="Superstrong approximation">Superstrong approximation</a></li> <li><a href="/wiki/Transfer_operator" title="Transfer operator">Transfer operator</a></li> <li><a href="/wiki/Transform_theory" title="Transform theory">Transform theory</a></li> <li><a href="/wiki/Weyl_law" title="Weyl law">Weyl law</a></li> <li><a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Self-adjoint&oldid=1236417132">https://en.wikipedia.org/w/index.php?title=Self-adjoint&oldid=1236417132</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" 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