Factorized sectorial relations, their maximal sectorial extensions, and form sums

Abstract

In this paper sectorial operators, or more generally, sectorial relations and their maximal sectorial extensions in a Hilbert space $\mathfrak{H}$ are considered. The particular interest is in sectorial relations $S$, which can be expressed in the factorized form \[ S=T^*(I+iB)T \quad \text{or} \quad S=T(I+iB)T^*, \] where $B$ is a bounded selfadjoint operator in a Hilbert space $\mathfrak{K}$ and $T:\mathfrak{H}\to \mathfrak{K}$ or $T:\mathfrak{K}\to \mathfrak{H}$, respectively, is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of $S$, a description of all the maximal sectorial extensions of $S$ is given with a straightforward construction of the extreme extensions $S_F$, the Friedrichs extension, and $S_K$, the Kre\uın extension of $S$, which uses the above factorized form of $S$. As an application of this construction the form sum of maximal sectorial extensions of two sectorial relations is treated.

24 pages. To appear in Banach Journal of Mathematical Analysis