View source for Self-adjoint operator

''Hermitian operator''

A [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838901.png" /> defined on a linear everywhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838902.png" /> in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838903.png" /> and coinciding with its [[Adjoint operator|adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838904.png" />, that is, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838905.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838907.png" />. Every self-adjoint operator is closed and cannot be extended with the preservation of (*) to a linear manifold wider than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838908.png" />; in view of this a self-adjoint operator is also called hypermaximal. Therefore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s0838909.png" /> is a bounded self-adjoint operator, then it is defined on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389010.png" />.

Every self-adjoint operator uniquely determines a [[Resolution of the identity|resolution of the identity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389012.png" />; the following integral representation holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389013.png" /></td> </tr></table>

where the integral is understood as the strong limit of the integral sums for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389014.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389015.png" /></td> </tr></table>

The spectrum of a self-adjoint operator is non-empty and lies on the real line. The quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389016.png" /> generated by a self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083890/s08389017.png" /> is real, and this enables one to introduce the concept of a [[Positive operator|positive operator]].

Many boundary value problems of mathematical physics are described by means of self-adjoint operators.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elements of functional analysis" , Wiley  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  F. Riesz,   B. Szökefalvi-Nagy,   "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>



====Comments====
Cf. also [[Hermitian operator|Hermitian operator]]; [[Symmetric operator|Symmetric operator]]; [[Self-adjoint linear transformation|Self-adjoint linear transformation]].