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

Abstract

In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space $\mathfrak{H}$. Our particular interest is in sectorial relations $S$, which can be expressed in the factorized form $$S=T^{\ast }(I+iB)T\text{or}S=T(I+iB)T^{\ast },$$ where $B$ is a bounded self-adjoint 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, along with a straightforward construction of the extreme extensions $S_F$, the Friedrichs extension, and $S_K$, the Kreĭn extension of $S$, which uses the above factorized form of $S$. As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.