Namespaces
Variants
Actions

Difference between revisions of "Semi-bounded operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
−
A [[Symmetric operator|symmetric operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839601.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839602.png" /> for which there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839603.png" /> such that
+
{{TEX|done}}
 +
A [[Symmetric operator|symmetric operator]] $S$ on a Hilbert space $H$ for which there exists a number $c$ such that
  
−
<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/s083960/s0839604.png" /></td> </tr></table>
+
$$(Sx,x)\geq c(x,x)$$
  
−
for all vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839605.png" /> in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839606.png" />. A semi-bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839607.png" /> always has a semi-bounded self-adjoint extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839608.png" /> with the same lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s0839609.png" /> (Friedrichs' theorem). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083960/s08396010.png" /> and its extension have the same deficiency indices (cf. [[Defective value|Defective value]]).
+
for all vectors $x$ in the domain of definition of $S$. A semi-bounded operator $S$ always has a semi-bounded self-adjoint extension $A$ with the same lower bound $c$ (Friedrichs' theorem). In particular, $S$ and its extension have the same deficiency indices (cf. [[Defective value|Defective value]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>

Latest revision as of 17:01, 2 July 2014

A symmetric operator $S$ on a Hilbert space $H$ for which there exists a number $c$ such that

$$(Sx,x)\geq c(x,x)$$

for all vectors $x$ in the domain of definition of $S$. A semi-bounded operator $S$ always has a semi-bounded self-adjoint extension $A$ with the same lower bound $c$ (Friedrichs' theorem). In particular, $S$ and its extension have the same deficiency indices (cf. Defective value).

References

[1] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
How to Cite This Entry:
Semi-bounded operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-bounded_operator&oldid=16008
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Semi-bounded_operator&oldid=32353"