SherbyNet - /page/wikibook/entry.mu

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Positive operator

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In <a href="Mathematics" title="Mathematics">mathematics</a> (specifically <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="Operator_theory" title="Operator theory">operator theory</a>, and <a href="Functional_analysis" title="Functional analysis">functional analysis</a>) as well as <a href="Physics" title="Physics">physics</a>, a <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> <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><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"> acting on an <a href="Inner_product_space" title="Inner product space">inner product space</a> is called positive-semidefinite (or non-negative) if, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {Dom} (A)}">

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

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</math><img src="./a9e59beffee590e83cc3d781f1f722e9b945d89b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.597ex; height:2.843ex;" alt="{\displaystyle x\in \operatorname {Dom} (A)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,x\rangle \in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \langle Ax,x\rangle \in \mathbb {R} }</annotation>

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</math><img src="./93608a6c480a686026b9d54639f8c24f3d49151b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.764ex; height:2.843ex;" alt="{\displaystyle \langle Ax,x\rangle \in \mathbb {R} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,x\rangle \geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \langle Ax,x\rangle \geq 0}</annotation>

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</math><img src="./39adffea6570e3b211ff26e644b2dc82ffc2eec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.507ex; height:2.843ex;" alt="{\displaystyle \langle Ax,x\rangle \geq 0}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dom} (A)}">

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

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</math><img src="./e8b5155610b6829a1b4161234ee2b6a5ce411515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.426ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dom} (A)}" loading="lazy"> is the <a href="Domain_of_a_function" title="Domain of a function">domain</a> of <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><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">. Positive-semidefinite operators are denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\geq 0}">

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

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</math><img src="./ebee3e714a32508aa26658d5c88f6a09237c82dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.004ex; height:2.343ex;" alt="{\displaystyle A\geq 0}" loading="lazy">. The operator is said to be positive-definite, and written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A>0}">

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

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</math><img src="./232a94dbd97f2ce670d4987c5ad8ad82b072861b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle A>0}" loading="lazy">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,x\rangle >0}">

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

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</math><img src="./1c33b71399c15b8cd4725870c7845f185f63c45f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.507ex; height:2.843ex;" alt="{\displaystyle \langle Ax,x\rangle >0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathop {\mathrm {Dom} } (A)\setminus \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathop {\mathrm {Dom} } (A)\setminus \{0\}}</annotation>

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</math><img src="./e507443a970d8b5a6cd2f095a40614b7db5c684c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.279ex; height:2.843ex;" alt="{\displaystyle x\in \mathop {\mathrm {Dom} } (A)\setminus \{0\}}" loading="lazy">.<a href="#cite_note-1">[1]</a>

Many authors define a positive operator <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><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"> to be a <a href="Self-adjoint_operator" title="Self-adjoint operator">self-adjoint</a> (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not imply self adjointness.

In physics (specifically <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>), such operators represent <a href="Quantum_state" title="Quantum state">quantum states</a>, via the <a href="Density_matrix" title="Density matrix">density matrix</a> formalism.

<div class="mw-heading mw-heading2"><h2 id="Cauchy–Schwarz_inequality">Cauchy–Schwarz inequality</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></div>

Take the inner product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

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

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</math><img src="./9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"> to be <a href="Antilinear_map" title="Antilinear map">anti-linear</a> on the first argument and linear on the second and suppose that <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><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"> is positive and symmetric, the latter meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }</annotation>

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</math><img src="./3b148dd9277ab2b18762deba9d034b58832bb025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.241ex; height:2.843ex;" alt="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }" loading="lazy">.

Then the non negativity of

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle A(\lambda x+\mu y),\lambda x+\mu y\rangle =|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}\langle Ay,x\rangle +|\mu |^{2}\langle Ay,y\rangle \\[1mm]=|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}(\langle Ax,y\rangle )^{*}+|\mu |^{2}\langle Ay,y\rangle \end{aligned}}}</annotation>

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</math><img src="./dfa6fe16545b068eb9ae69e1de9b19ffd2883b04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.293ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}\langle A(\lambda x+\mu y),\lambda x+\mu y\rangle =|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}\langle Ay,x\rangle +|\mu |^{2}\langle Ay,y\rangle \\[1mm]=|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}(\langle Ax,y\rangle )^{*}+|\mu |^{2}\langle Ay,y\rangle \end{aligned}}}" loading="lazy"></dd></dl>

for all complex <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><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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> shows that

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<annotation encoding="application/x-tex">{\displaystyle \left|\langle Ax,y\rangle \right|^{2}\leq \langle Ax,x\rangle \langle Ay,y\rangle .}</annotation>

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</math><img src="./3a1a7cb26194214f7cef753559ad03c8d439c807.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.308ex; height:3.343ex;" alt="{\displaystyle \left|\langle Ax,y\rangle \right|^{2}\leq \langle Ax,x\rangle \langle Ay,y\rangle .}" loading="lazy"></dd></dl>

It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathop {\text{Im}} A\perp \mathop {\text{Ker}} A.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathop {\text{Im}} A\perp \mathop {\text{Ker}} A.}</annotation>

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</math><img src="./ee0a9d34d4515ffa10a9dec9c0a0dc92e4813b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.533ex; height:2.176ex;" alt="{\displaystyle \mathop {\text{Im}} A\perp \mathop {\text{Ker}} A.}" loading="lazy"> If <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><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"> is defined everywhere, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,x\rangle =0,}">

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

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</math><img src="./1ffd6449800e86b7cb990dcc692bc83a0e94be6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.153ex; height:2.843ex;" alt="{\displaystyle \langle Ax,x\rangle =0,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=0.}">

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

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</math><img src="./946fe81ea7ee3f2fc670d26260a9b2f2ba4933d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.981ex; height:2.176ex;" alt="{\displaystyle Ax=0.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="On_a_complex_Hilbert_space,_if_an_operator_is_non-negative_then_it_is_symmetric">On a complex Hilbert space, if an operator is non-negative then it is symmetric</h2></div>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \operatorname {Dom} A,}">

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<annotation encoding="application/x-tex">{\displaystyle x,y\in \operatorname {Dom} A,}</annotation>

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</math><img src="./e5923622dd858fa1d3e3146c2c36e5c0ec810124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.011ex; height:2.509ex;" alt="{\displaystyle x,y\in \operatorname {Dom} A,}" loading="lazy"> the <a href="Polarization_identity" title="Polarization identity">polarization identity</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle Ax,y\rangle ={\frac {1}{4}}({}&\langle A(x+y),x+y\rangle -\langle A(x-y),x-y\rangle \\[1mm]&{}-i\langle A(x+iy),x+iy\rangle +i\langle A(x-iy),x-iy\rangle )\end{aligned}}}</annotation>

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</math><img src="./8723ccc4576e55f1744625b2c81b2f99f465f3c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.688ex; margin-bottom: -0.316ex; width:59.318ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\langle Ax,y\rangle ={\frac {1}{4}}({}&\langle A(x+y),x+y\rangle -\langle A(x-y),x-y\rangle \\[1mm]&{}-i\langle A(x+iy),x+iy\rangle +i\langle A(x-iy),x-iy\rangle )\end{aligned}}}" loading="lazy"></dd></dl>

and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,x\rangle =\langle x,Ax\rangle ,}">

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<annotation encoding="application/x-tex">{\displaystyle \langle Ax,x\rangle =\langle x,Ax\rangle ,}</annotation>

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</math><img src="./75a8fdea78624d21bd86025f0b0e3a8e99cd464b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.237ex; height:2.843ex;" alt="{\displaystyle \langle Ax,x\rangle =\langle x,Ax\rangle ,}" loading="lazy"> for positive operators, show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,}">

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<annotation encoding="application/x-tex">{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,}</annotation>

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</math><img src="./19229dc754b0b52b2e040439244d9654709ead5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.888ex; height:2.843ex;" alt="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,}" loading="lazy"> so <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><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"> is symmetric.

In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbb {R} }}">

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<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {R} }}</annotation>

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</math><img src="./b4594f07af31202e24d1c21ae0c8bb424789df37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:2.509ex;" alt="{\displaystyle H_{\mathbb {R} }}" loading="lazy"> may not be symmetric. As a counterexample, define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation>

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</math><img src="./4f8fc2bca996c17b35ed48d71345b730ba057399.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.759ex; height:2.676ex;" alt="{\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}" loading="lazy"> to be an operator of rotation by an <a href="Acute_angle" class="mw-redirect" title="Acute angle">acute angle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in (-\pi /2,\pi /2).}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \in (-\pi /2,\pi /2).}</annotation>

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</math><img src="./ec7dea7920f0b190201b2aa39525b9b0bf822684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.973ex; height:2.843ex;" alt="{\displaystyle \varphi \in (-\pi /2,\pi /2).}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,x\rangle =\|Ax\|\|x\|\cos \varphi >0,}">

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<annotation encoding="application/x-tex">{\displaystyle \langle Ax,x\rangle =\|Ax\|\|x\|\cos \varphi >0,}</annotation>

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</math><img src="./cbef0503d64919159be1bb5f00fd2558e7f9e0a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.71ex; height:2.843ex;" alt="{\displaystyle \langle Ax,x\rangle =\|Ax\|\|x\|\cos \varphi >0,}" loading="lazy"> but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}=A^{-1}\neq A,}">

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<annotation encoding="application/x-tex">{\displaystyle A^{*}=A^{-1}\neq A,}</annotation>

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</math><img src="./3b9d03b42e51ae459b0184e91f5fde9e6258e848.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.46ex; height:3.176ex;" alt="{\displaystyle A^{*}=A^{-1}\neq A,}" loading="lazy"> so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math><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"> is not symmetric.

<div class="mw-heading mw-heading2"><h2 id="If_an_operator_is_non-negative_and_defined_on_the_whole_complex_Hilbert_space,_then_it_is_self-adjoint_and_bounded">If an operator is non-negative and defined on the whole complex Hilbert space, then it is self-adjoint and <a href="Bounded_operator" title="Bounded operator">bounded</a></h2></div>

The symmetry of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dom} A\subseteq \operatorname {Dom} A^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Dom} A\subseteq \operatorname {Dom} A^{*}}</annotation>

</semantics>

</math><img src="./12d1f93420c3c22cdac36c7971a2fa42583c54d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.161ex; height:2.509ex;" alt="{\displaystyle \operatorname {Dom} A\subseteq \operatorname {Dom} A^{*}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A^{*}|_{\operatorname {Dom} (A)}.}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=A^{*}|_{\operatorname {Dom} (A)}.}</annotation>

</semantics>

</math><img src="./58a2d3eeee0498eb14264867cc8019c52c50e789.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.123ex; height:3.343ex;" alt="{\displaystyle A=A^{*}|_{\operatorname {Dom} (A)}.}" loading="lazy"> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> to be self-adjoint, it is necessary that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dom} A=\operatorname {Dom} A^{*}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Dom} A=\operatorname {Dom} A^{*}.}</annotation>

</semantics>

</math><img src="./848da382001cdc2227530526c81f254eba53671b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.808ex; height:2.343ex;" alt="{\displaystyle \operatorname {Dom} A=\operatorname {Dom} A^{*}.}" loading="lazy"> In our case, the equality of <a href="Domain_of_a_function" title="Domain of a function">domains</a> holds because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbb {C} }=\operatorname {Dom} A\subseteq \operatorname {Dom} A^{*},}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {C} }=\operatorname {Dom} A\subseteq \operatorname {Dom} A^{*},}</annotation>

</semantics>

</math><img src="./293ff246d3897e8f971ee77d0248195fc088944e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.256ex; height:2.676ex;" alt="{\displaystyle H_{\mathbb {C} }=\operatorname {Dom} A\subseteq \operatorname {Dom} A^{*},}" loading="lazy"> so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is indeed self-adjoint. The fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is bounded now follows from the <a href="Hellinger%E2%80%93Toeplitz_theorem" title="Hellinger–Toeplitz theorem">Hellinger–Toeplitz theorem</a>.

This property does not hold on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbb {R} }.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {R} }.}</annotation>

</semantics>

</math><img src="./5c12ae147cc136c8f03d6460f659dd2df6cbfd0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.997ex; height:2.509ex;" alt="{\displaystyle H_{\mathbb {R} }.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Partial_order_of_self-adjoint_operators">Partial order of self-adjoint operators</h2></div>

A natural <a href="Partial_ordering" class="mw-redirect" title="Partial ordering">partial ordering</a> of self-adjoint operators arises from the definition of positive operators. Define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\geq A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ccfba23ef8e7e0ddb0fc0282dbf2cbc38148f9cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\geq A}" loading="lazy"> if the following hold:

</mstyle>

</mrow>

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

</semantics>

</math><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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> are self-adjoint</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b82610462af3b50f2b81f9cfab48b1f30fd25047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.608ex; height:2.343ex;" alt="{\displaystyle B-A\geq 0}" loading="lazy"></li></ol>

It can be seen that a similar result as the <a href="Monotone_convergence_theorem" title="Monotone convergence theorem">Monotone convergence theorem</a> holds for <a href="Monotone_increasing" class="mw-redirect" title="Monotone increasing">monotone increasing</a>, bounded, self-adjoint operators on Hilbert spaces.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Application_to_physics:_quantum_states">Application to physics: quantum states</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Quantum_state" title="Quantum state">Quantum state</a> and <a href="Density_operator" class="mw-redirect" title="Density operator">Density operator</a></div>

The definition of a <a href="Quantum_system" class="mw-redirect" title="Quantum system">quantum system</a> includes a complex <a href="Separable_Hilbert_space" class="mw-redirect" title="Separable Hilbert space">separable Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbb {C} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {C} }}</annotation>

</semantics>

</math><img src="./7a86593eab9a6ea3393e69253723d10a7da7fc2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:2.509ex;" alt="{\displaystyle H_{\mathbb {C} }}" loading="lazy"> and a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> of positive <a href="Trace-class" class="mw-redirect" title="Trace-class">trace-class</a> <a href="Density_operator" class="mw-redirect" title="Density operator">operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbb {C} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {C} }}</annotation>

</semantics>

</math><img src="./7a86593eab9a6ea3393e69253723d10a7da7fc2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:2.509ex;" alt="{\displaystyle H_{\mathbb {C} }}" loading="lazy"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathop {\text{Trace}} \rho =1.}">

<mtext>Trace</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathop {\text{Trace}} \rho =1.}</annotation>

</semantics>

</math><img src="./58b736be65afff170e5df21bcd060bfde3882fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.314ex; height:2.676ex;" alt="{\displaystyle \mathop {\text{Trace}} \rho =1.}" loading="lazy"> The <a href="Set_(mathematics)" title="Set (mathematics)">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> is the set of states. Every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \in {\cal {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5a80648e1c83f7da911cab0f2a844d2693ed376f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.535ex; height:2.676ex;" alt="{\displaystyle \rho \in {\cal {S}}}" loading="lazy"> is called a state or a density operator. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \in H_{\mathbb {C} },}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \in H_{\mathbb {C} },}</annotation>

</semantics>

</math><img src="./88faad088d317774a7626ced47a946720557a164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.351ex; height:2.509ex;" alt="{\displaystyle \psi \in H_{\mathbb {C} },}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\psi \|=1,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\psi \|=1,}</annotation>

</semantics>

</math><img src="./674b74d77fa9bea7d05920eb5020ca347101f067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.746ex; height:2.843ex;" alt="{\displaystyle \|\psi \|=1,}" loading="lazy"> the operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\psi }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./513dbe115912bf35eb11e964b02eb63d1a5fb6b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.794ex; height:2.843ex;" alt="{\displaystyle P_{\psi }}" loading="lazy"> of projection onto the <a href="Linear_span" title="Linear span">span</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> is called a <a href="Pure_state" class="mw-redirect" title="Pure state">pure state</a>. (Since each pure state is identifiable with a <a href="Unit_vector" title="Unit vector">unit vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \in H_{\mathbb {C} },}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \in H_{\mathbb {C} },}</annotation>

</semantics>

</math><img src="./88faad088d317774a7626ced47a946720557a164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.351ex; height:2.509ex;" alt="{\displaystyle \psi \in H_{\mathbb {C} },}" loading="lazy"> some sources define pure states to be unit elements from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbb {C} }).}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {C} }).}</annotation>

</semantics>

</math><img src="./cca81aceaa1b28104b1ec62170e2130b03dd6f26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.902ex; height:2.843ex;" alt="{\displaystyle H_{\mathbb {C} }).}" loading="lazy"> States that are not pure are called <a href="Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed</a>.

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

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

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFRoman2008">Roman 2008</a>, p. 250 §10

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.

</li>

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

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

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</style><cite id="CITEREFConway1990" class="citation cs2">Conway, John B. (1990), Functional Analysis: An Introduction, <a href="Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-97245-5</bdi></cite></li>

<li><cite id="CITEREFRoman2008" class="citation cs2"><a href="Steven_Roman" title="Steven Roman">Roman, Stephen</a> (2008), Advanced Linear Algebra, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> (Third ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-72828-5</bdi></cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-18" href="https://en.wikipedia.org/wiki/?title=Positive_operator&oldid=1301182795">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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