# Semi-bounded operator

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).