View source for Semi-bounded operator

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

<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>

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]]).

====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>