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

In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space H. Our particular interest is in sectorial relations S, which can be expressed in the factorized form

S = T* (I + iB)T or S = T (I + iB)T*;

where B is a bounded self-adjoint operator in a Hilbert space K and T : H -> K (or T : K -> H, respectively) is a linear operator or a linear relation which is not assumed to be closed. Using the speci fi c 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 Krein 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.