Namespaces
Variants
Actions

Difference between revisions of "Self-adjoint operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
s0838901.png
 +
$#A+1 = 17 n = 0
 +
$#C+1 = 17 : ~/encyclopedia/old_files/data/S083/S.0803890 Self\AAhadjoint operator,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''Hermitian 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
+
A [[Linear operator|linear operator]] $  A $
 +
defined on a linear everywhere-dense set $  D ( A) $
 +
in a Hilbert space $  H $
 +
and coinciding with its [[Adjoint operator|adjoint operator]] $  A  ^ {*} $,  
 +
that is, such that $  D ( A) = D ( A  ^ {*} ) $
 +
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>
+
$$ \tag{* }
 +
\langle  Ax, y \rangle  = \langle  x, Ay \rangle
 +
$$
  
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" />.
+
for every $  x, y \in D ( A) $.  
 +
Every self-adjoint operator is closed and cannot be extended with the preservation of (*) to a linear manifold wider than $  D ( A) $;  
 +
in view of this a self-adjoint operator is also called hypermaximal. Therefore, if $  A $
 +
is a bounded self-adjoint operator, then it is defined on the whole of $  H $.
  
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:
+
Every self-adjoint operator uniquely determines a [[Resolution of the identity|resolution of the identity]] $  E _  \lambda  $,
 +
$  - \infty < \lambda < \infty $;  
 +
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>
+
$$
 +
Ax  = \int\limits _ {- \infty } ^  \infty  \lambda  dE _  \lambda  x,
 +
$$
  
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
+
where the integral is understood as the strong limit of the integral sums for each $  x \in D ( A) $,  
 +
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>
+
$$
 +
D ( A)  = \
 +
\left \{ {x } : {
 +
\int\limits _ {- \infty } ^  \infty 
 +
\lambda  ^ {2}  d \langle  E _  \lambda  x, x \rangle < \infty
 +
} \right \}
 +
.
 +
$$
  
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]].
+
The spectrum of a self-adjoint operator is non-empty and lies on the real line. The quadratic form $  K ( A) = \langle  Ax, x \rangle $
 +
generated by a self-adjoint operator $  A $
 +
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.
 
Many boundary value problems of mathematical physics are described by means of self-adjoint operators.
Line 21: Line 57:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
Cf. also [[Hermitian operator|Hermitian operator]]; [[Symmetric operator|Symmetric operator]]; [[Self-adjoint linear transformation|Self-adjoint linear transformation]].
 
Cf. also [[Hermitian operator|Hermitian operator]]; [[Symmetric operator|Symmetric operator]]; [[Self-adjoint linear transformation|Self-adjoint linear transformation]].

Latest revision as of 08:13, 6 June 2020


Hermitian operator

A linear operator $ A $ defined on a linear everywhere-dense set $ D ( A) $ in a Hilbert space $ H $ and coinciding with its adjoint operator $ A ^ {*} $, that is, such that $ D ( A) = D ( A ^ {*} ) $ and

$$ \tag{* } \langle Ax, y \rangle = \langle x, Ay \rangle $$

for every $ x, y \in D ( A) $. Every self-adjoint operator is closed and cannot be extended with the preservation of (*) to a linear manifold wider than $ D ( A) $; in view of this a self-adjoint operator is also called hypermaximal. Therefore, if $ A $ is a bounded self-adjoint operator, then it is defined on the whole of $ H $.

Every self-adjoint operator uniquely determines a resolution of the identity $ E _ \lambda $, $ - \infty < \lambda < \infty $; the following integral representation holds:

$$ Ax = \int\limits _ {- \infty } ^ \infty \lambda dE _ \lambda x, $$

where the integral is understood as the strong limit of the integral sums for each $ x \in D ( A) $, and

$$ D ( A) = \ \left \{ {x } : { \int\limits _ {- \infty } ^ \infty \lambda ^ {2} d \langle E _ \lambda x, x \rangle < \infty } \right \} . $$

The spectrum of a self-adjoint operator is non-empty and lies on the real line. The quadratic form $ K ( A) = \langle Ax, x \rangle $ generated by a self-adjoint operator $ A $ is real, and this enables one to introduce the concept of a positive operator.

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

References

[1] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Wiley (1974) (Translated from Russian)
[2] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[3] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)

Comments

Cf. also Hermitian operator; Symmetric operator; Self-adjoint linear transformation.

How to Cite This Entry:
Self-adjoint operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-adjoint_operator&oldid=48649
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article