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Difference between revisions of "Semi-bounded operator"

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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
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A [[Symmetric operator|symmetric operator]] $S$ on a Hilbert space $H$ for which there exists a number $c$ such that
  
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<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>
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$$(Sx,x)\geq c(x,x)$$
  
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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]]).
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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=32353
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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