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)TorS=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 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 SF, the Friedrichs extension, and SK, 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.