View source for Hermitian operator

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''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====
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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>
<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>