# Semi-bounded operator

From Encyclopedia of Mathematics

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://www.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