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Difference between revisions of "Hermitian operator"

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''symmetric operator''
 
''symmetric operator''
  
A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470901.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470902.png" /> with a dense domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470903.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470904.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470905.png" />. This condition is equivalent to: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470906.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470907.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470908.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h0470909.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h04709010.png" />, that is, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h04709011.png" />. A bounded Hermitian operator is either defined on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h04709012.png" /> or can be so extended by continuity, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h04709013.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047090/h04709014.png" /> is a [[Self-adjoint operator|self-adjoint operator]]. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a [[Hermitian matrix|Hermitian matrix]].
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A linear operator $A$ on a Hilbert space $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ is a [[self-adjoint operator]]. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a [[Hermitian matrix]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR>
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</table>

Revision as of 20:31, 26 April 2016


symmetric operator

A linear operator $A$ on a Hilbert space $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ is a self-adjoint operator. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a Hermitian matrix.

References

[1] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[2] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
How to Cite This Entry:
Hermitian operator. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hermitian_operator&oldid=29566
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article