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

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<p>In mathematics, specifically in <a href="Spectral_theory" title="Spectral theory">spectral theory</a>, an <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> of a <a href="Unbounded_operator#Closed_linear_operators" title="Unbounded operator">closed linear operator</a> is called <b>normal</b> if the space admits a decomposition into a direct sum of a finite-dimensional <a href="Generalized_eigenspace" class="mw-redirect" title="Generalized eigenspace">generalized eigenspace</a> and an <a href="Invariant_subspace" title="Invariant subspace">invariant subspace</a> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I}">

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

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</math></span><img src="./2ba67e098a60d3285b10f2e76be219fc3c52d690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.11ex; height:2.343ex;" alt="{\displaystyle A-\lambda I}" loading="lazy"></span> has a bounded inverse.

The set of normal eigenvalues coincides with the <a href="Discrete_spectrum_(mathematics)" title="Discrete spectrum (mathematics)">discrete spectrum</a>.

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<div class="mw-heading mw-heading2"><h2 id="Root_lineal">Root lineal</h2></div>

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

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</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span> be a <a href="Banach_space" title="Banach space">Banach space</a>. The <a href="Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space" title="Generalized eigenvector">root lineal</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {L}}_{\lambda }(A)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}_{\lambda }(A)}</annotation>

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</math></span><img src="./9f1a6ebd340867e2ad8311a99fb2948ae67ce7d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {L}}_{\lambda }(A)}" loading="lazy"></span> of a linear operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}">

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<annotation encoding="application/x-tex">{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}</annotation>

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</math></span><img src="./902778710bb8968f0662940c2e223f3bd2e389ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.79ex; height:2.176ex;" alt="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}" loading="lazy"></span> with domain <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {D}}(A)}">

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

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</math></span><img src="./5b03f2944633b56142fefe09b82d8e50e92315f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.486ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {D}}(A)}" loading="lazy"></span> corresponding to the eigenvalue <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \sigma _{p}(A)}">

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

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</math></span><img src="./c209ce9c4c49a1a7bda9fdfe072da00b457f29e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.135ex; height:3.009ex;" alt="{\displaystyle \lambda \in \sigma _{p}(A)}" loading="lazy"></span> is defined as

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}_{\lambda }(A)=\bigcup _{k\in \mathbb {N} }\{x\in {\mathfrak {D}}(A):\,(A-\lambda I_{\mathfrak {B}})^{j}x\in {\mathfrak {D}}(A)\,\forall j\in \mathbb {N} ,\,j\leq k;\,(A-\lambda I_{\mathfrak {B}})^{k}x=0\}\subset {\mathfrak {B}},}</annotation>

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</math></span><img src="./0dba536ab3501d8172f1e9eedebab91fa1598b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:86.057ex; height:5.676ex;" alt="{\displaystyle {\mathfrak {L}}_{\lambda }(A)=\bigcup _{k\in \mathbb {N} }\{x\in {\mathfrak {D}}(A):\,(A-\lambda I_{\mathfrak {B}})^{j}x\in {\mathfrak {D}}(A)\,\forall j\in \mathbb {N} ,\,j\leq k;\,(A-\lambda I_{\mathfrak {B}})^{k}x=0\}\subset {\mathfrak {B}},}" loading="lazy"></span></dd></dl>

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

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

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</math></span><img src="./d19db13f4f526fe6dc93377b2758a41d56916e35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.708ex; height:2.509ex;" alt="{\displaystyle I_{\mathfrak {B}}}" loading="lazy"></span> is the identity operator in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}">

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

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</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span>.

This set is a <a href="Linear_manifold" class="mw-redirect" title="Linear manifold">linear manifold</a> but not necessarily a <a href="Vector_space" title="Vector space">vector space</a>, since it is not necessarily closed in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}">

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

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</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span>. If this set is closed (for example, when it is finite-dimensional), it is called the <a href="Generalized_eigenspace" class="mw-redirect" title="Generalized eigenspace">generalized eigenspace</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> corresponding to the eigenvalue <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Definition_of_a_normal_eigenvalue">Definition of a normal eigenvalue</h2></div>

<p>An <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \sigma _{p}(A)}">

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

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</math></span><img src="./c209ce9c4c49a1a7bda9fdfe072da00b457f29e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.135ex; height:3.009ex;" alt="{\displaystyle \lambda \in \sigma _{p}(A)}" loading="lazy"></span> of a <a href="Unbounded_operator#Closed_linear_operators" title="Unbounded operator">closed linear operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}">

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<annotation encoding="application/x-tex">{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}</annotation>

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</math></span><img src="./902778710bb8968f0662940c2e223f3bd2e389ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.79ex; height:2.176ex;" alt="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}" loading="lazy"></span> in the <a href="Banach_space" title="Banach space">Banach space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}">

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

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</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span> with <a href="Unbounded_operator#Definitions_and_basic_properties" title="Unbounded operator">domain</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}}">

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

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</math></span><img src="./a401d9ace0d9b1387a08c2e270e0b0054c93dcd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.639ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}}" loading="lazy"></span> is called <i>normal</i> (in the original terminology, <i><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> corresponds to a normally splitting finite-dimensional root subspace</i>), if the following two conditions are satisfied:

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<ol><li>The <a href="Algebraic_multiplicity" class="mw-redirect" title="Algebraic multiplicity">algebraic multiplicity</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> is finite: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)<\infty }">

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<annotation encoding="application/x-tex">{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)<\infty }</annotation>

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</math></span><img src="./7f67f842d85981fd3982d7650a48a6d8356193ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.306ex; height:2.843ex;" alt="{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)<\infty }" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {L}}_{\lambda }(A)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}_{\lambda }(A)}</annotation>

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</math></span><img src="./9f1a6ebd340867e2ad8311a99fb2948ae67ce7d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {L}}_{\lambda }(A)}" loading="lazy"></span> is the <a href="Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space" title="Generalized eigenvector">root lineal</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> corresponding to the eigenvalue <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span>;</li>

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

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

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</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span> could be decomposed into a direct sum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }}</annotation>

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</math></span><img src="./108d33e5fc3b2896dd97dade59316891f9ac1edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.408ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {B}}={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{\lambda }}">

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

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</math></span><img src="./2c61e3195d9428eb4e8b351981f0d6e6abfcfe19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.124ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {N}}_{\lambda }}" loading="lazy"></span> is an <a href="Invariant_subspace" title="Invariant subspace">invariant subspace</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> in which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I_{\mathfrak {B}}}">

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

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</math></span><img src="./48a4d189209c51a6b0bf43774ffcecb6a011b3c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.647ex; height:2.509ex;" alt="{\displaystyle A-\lambda I_{\mathfrak {B}}}" loading="lazy"></span> has a bounded inverse.</li></ol>

<p>That is, the restriction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

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</math></span><img src="./3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> onto <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{\lambda }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2c61e3195d9428eb4e8b351981f0d6e6abfcfe19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.124ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {N}}_{\lambda }}" loading="lazy"></span> is an operator with domain <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {D}}(A_{2})={\mathfrak {N}}_{\lambda }\cap {\mathfrak {D}}(A)}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {D}}(A_{2})={\mathfrak {N}}_{\lambda }\cap {\mathfrak {D}}(A)}</annotation>

</semantics>

</math></span><img src="./7c2157494d446e44e852fe55427ae3bf6db2e3da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.831ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {D}}(A_{2})={\mathfrak {N}}_{\lambda }\cap {\mathfrak {D}}(A)}" loading="lazy"></span> and with the range <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}(A_{2}-\lambda I)\subset {\mathfrak {N}}_{\lambda }}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}(A_{2}-\lambda I)\subset {\mathfrak {N}}_{\lambda }}</annotation>

</semantics>

</math></span><img src="./8fc9aa59ec7fdfd22c348b249fd1cba1838c792c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.121ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {R}}(A_{2}-\lambda I)\subset {\mathfrak {N}}_{\lambda }}" loading="lazy"></span> which has a bounded inverse.<sup id="cite_ref-gohberg1957_1-0" class="reference"><a href="#cite_note-gohberg1957-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gohberg1960_2-0" class="reference"><a href="#cite_note-gohberg1960-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gohberg1969_3-0" class="reference"><a href="#cite_note-gohberg1969-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Equivalent_characterizations_of_normal_eigenvalues">Equivalent characterizations of normal eigenvalues</h2></div>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}</annotation>

</semantics>

</math></span><img src="./902778710bb8968f0662940c2e223f3bd2e389ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.79ex; height:2.176ex;" alt="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}" loading="lazy"></span> be a closed linear <a href="Densely_defined_operator" title="Densely defined operator">densely defined operator</a> in the Banach space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span>. The following statements are equivalent<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>(Theorem III.88):

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba79162a628bddf53195751534366dd1282df423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma (A)}" loading="lazy"></span> is a normal eigenvalue;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba79162a628bddf53195751534366dd1282df423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma (A)}" loading="lazy"></span> is an isolated point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I_{\mathfrak {B}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48a4d189209c51a6b0bf43774ffcecb6a011b3c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.647ex; height:2.509ex;" alt="{\displaystyle A-\lambda I_{\mathfrak {B}}}" loading="lazy"></span> is <a href="Fredholm_operator#semi-Fredholm_operators" title="Fredholm operator">semi-Fredholm</a>;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba79162a628bddf53195751534366dd1282df423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma (A)}" loading="lazy"></span> is an isolated point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I_{\mathfrak {B}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48a4d189209c51a6b0bf43774ffcecb6a011b3c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.647ex; height:2.509ex;" alt="{\displaystyle A-\lambda I_{\mathfrak {B}}}" loading="lazy"></span> is <a href="Fredholm_operator" title="Fredholm operator">Fredholm</a>;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba79162a628bddf53195751534366dd1282df423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma (A)}" loading="lazy"></span> is an isolated point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I_{\mathfrak {B}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48a4d189209c51a6b0bf43774ffcecb6a011b3c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.647ex; height:2.509ex;" alt="{\displaystyle A-\lambda I_{\mathfrak {B}}}" loading="lazy"></span> is <a href="Fredholm_operator" title="Fredholm operator">Fredholm</a> of index zero;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba79162a628bddf53195751534366dd1282df423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma (A)}" loading="lazy"></span> is an isolated point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"></span> and the rank of the corresponding <a href="Riesz_projector" title="Riesz projector">Riesz projector</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./330591f9b6fffc93ca78514576fd0d8cfac6f0c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.683ex; height:2.509ex;" alt="{\displaystyle P_{\lambda }}" loading="lazy"></span> is finite;</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba79162a628bddf53195751534366dd1282df423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma (A)}" loading="lazy"></span> is an isolated point in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (A)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a507cac58b93e190d8f9ff7e470e6c28ff772d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \sigma (A)}" loading="lazy"></span>, its algebraic multiplicity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)}</annotation>

</semantics>

</math></span><img src="./5d0530dab6d6d3e33f4a37863912ee884e6a60c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.884ex; height:2.843ex;" alt="{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)}" loading="lazy"></span> is finite, and the range of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda I_{\mathfrak {B}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48a4d189209c51a6b0bf43774ffcecb6a011b3c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.647ex; height:2.509ex;" alt="{\displaystyle A-\lambda I_{\mathfrak {B}}}" loading="lazy"></span> is <a href="Closed_range_theorem" title="Closed range theorem">closed</a>.<sup id="cite_ref-gohberg1957_1-1" class="reference"><a href="#cite_note-gohberg1957-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gohberg1960_2-1" class="reference"><a href="#cite_note-gohberg1960-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gohberg1969_3-1" class="reference"><a href="#cite_note-gohberg1969-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ol>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> is a normal eigenvalue, then the root lineal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {L}}_{\lambda }(A)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}_{\lambda }(A)}</annotation>

</semantics>

</math></span><img src="./9f1a6ebd340867e2ad8311a99fb2948ae67ce7d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {L}}_{\lambda }(A)}" loading="lazy"></span> coincides with the range of the Riesz projector, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}(P_{\lambda })}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}(P_{\lambda })}</annotation>

</semantics>

</math></span><img src="./8c89c0fbd79f51c9588c384419544e3bfc4afbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.416ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {R}}(P_{\lambda })}" loading="lazy"></span>.<sup id="cite_ref-gohberg1969_3-2" class="reference"><a href="#cite_note-gohberg1969-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_the_discrete_spectrum">Relation to the discrete spectrum</h2></div>

<p>The above equivalence shows that the set of normal eigenvalues coincides with the <a href="Discrete_spectrum_(Mathematics)" class="mw-redirect" title="Discrete spectrum (Mathematics)">discrete spectrum</a>, defined as the set of isolated points of the spectrum with finite rank of the corresponding Riesz projector.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Decomposition_of_the_spectrum_of_nonselfadjoint_operators">Decomposition of the spectrum of nonselfadjoint operators</h2></div>

<p>The spectrum of a closed operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}</annotation>

</semantics>

</math></span><img src="./902778710bb8968f0662940c2e223f3bd2e389ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.79ex; height:2.176ex;" alt="{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}" loading="lazy"></span> in the Banach space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span> can be decomposed into the union of two disjoint sets, the set of normal eigenvalues and the fifth type of the <a href="Essential_spectrum" title="Essential spectrum">essential spectrum</a>:

</p>

<mtext>normal eigenvalues of</mtext>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (A)=\{{\text{normal eigenvalues of}}\ A\}\cup \sigma _{\mathrm {ess} ,5}(A).}</annotation>

</semantics>

</math></span><img src="./04cd1a0ce5eb84417c7d7078ecd52a823ddddb7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.868ex; height:3.009ex;" alt="{\displaystyle \sigma (A)=\{{\text{normal eigenvalues of}}\ A\}\cup \sigma _{\mathrm {ess} ,5}(A).}" loading="lazy"></span></dd></dl>

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

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

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

<li><a href="Fredholm_operator" title="Fredholm operator">Fredholm operator</a></li>

<li><a href="Operator_theory" title="Operator theory">Operator theory</a></li>

<li><a href="Resolvent_formalism" title="Resolvent formalism">Resolvent formalism</a></li>

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

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

<li><a href="Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">Spectrum of an operator</a></li></ul>

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

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</style><cite id="CITEREFGohberg,_I._CKreĭn,_M._G.1957" class="citation journal cs1">Gohberg, I. C; Kreĭn, M. G. (1957). <a rel="nofollow" class="external text" href="https://mi.mathnet.ru/umn7581">"Основные положения о дефектных числах, корневых числах и индексах линейных операторов"</a> [Fundamental aspects of defect numbers, root numbers and indexes of linear operators]. <i>Uspekhi Mat. Nauk</i> [<i>Amer. Math. Soc. Transl. (2)</i>]. New Series. <b>12</b> (2(74)): <span class="nowrap">43–</span>118.</cite></span>

</li>

<li id="cite_note-gohberg1960-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-gohberg1960_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gohberg1960_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGohberg,_I._CKreĭn,_M._G.1960" class="citation journal cs1">Gohberg, I. C; Kreĭn, M. G. (1960). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://mi.mathnet.ru/umn7581">"Fundamental aspects of defect numbers, root numbers and indexes of linear operators"</a></span>. <i>American Mathematical Society Translations</i>. <b>13</b>: <span class="nowrap">185–</span>264. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Ftrans2%2F013%2F08">10.1090/trans2/013/08</a>.</cite></span>

</li>

<li id="cite_note-gohberg1969-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-gohberg1969_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gohberg1969_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-gohberg1969_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGohberg,_I._CKreĭn,_M._G.1969" class="citation book cs1">Gohberg, I. C; Kreĭn, M. G. (1969). <a rel="nofollow" class="external text" href="http://gen.lib.rus.ec/book/index.php?md5=9CE2F03854312C3E29ED684CD84D8CA3"><i>Introduction to the theory of linear nonselfadjoint operators</i></a>. American Mathematical Society, Providence, R.I.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFBoussaid,_N.Comech,_A.2019" class="citation book cs1">Boussaid, N.; Comech, A. (2019). <a rel="nofollow" class="external text" href="https://bookstore.ams.org/surv-244"><i>Nonlinear Dirac equation. Spectral stability of solitary waves</i></a>. American Mathematical Society, Providence, R.I. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4704-4395-5</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFReed,_M.Simon,_B.1978" class="citation book cs1">Reed, M.; Simon, B. (1978). <i>Methods of modern mathematical physics, vol. IV. Analysis of operators</i>. Academic Press [Harcourt Brace Jovanovich Publishers], New York.</cite></span>

</li>

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

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</style><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach</a></li>

<li><a href="Besov_space" title="Besov space">Besov</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li>

<li><a href="Nuclear_space" title="Nuclear space">Nuclear</a></li>

<li><a href="Orlicz_space" title="Orlicz space">Orlicz</a></li>

<li><a href="Schwartz_space" title="Schwartz space">Schwartz</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</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="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Dual_space" title="Dual space">Dual</a> (<a href="Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li>

<li><a href="Reflexive_space" title="Reflexive space">Reflexive</a></li>

<li><a href="Separable_space" title="Separable space">Separable</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</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="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li>

<li><a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li>

<li><a href="Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li>

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

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</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="Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li>

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

<li><a href="Nuclear_operator" title="Nuclear operator">Nuclear</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li>

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">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="Banach_algebra" title="Banach algebra">Banach algebra</a></li>

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

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

<li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</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="Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li>

<li><a href="Mahler's_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</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-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hardy_space" title="Hardy space">Hardy space</a></li>

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

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

<li><a href="Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li>

<li><a href="Functional_calculus" title="Functional calculus">Functional calculus</a></li>

<li><a href="Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li>

<li><a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li>

<li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</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="Approximation_property" title="Approximation property">Approximation property</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced set</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Weak_topology" title="Weak topology">Weak topology</a></li>

<li><a href="Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li>

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

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Spectral_theory_and_*-algebras154" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Spectral_theory_and_*-algebras154" style="font-size:114%;margin:0 4em"><a href="Spectral_theory" title="Spectral theory">Spectral theory</a> and <a href="*-algebra" title="*-algebra"><sup>*</sup>-algebras</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This article is issued from <a class="external text" title="Last edited on 2025-05-27" href="https://en.wikipedia.org/wiki/?title=Normal_eigenvalue&oldid=1292604434">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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