Semi-bounded operator

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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).


[1] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
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Semi-bounded operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article