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

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 $$S=T^{\ast }(I+iB)T\text{or}S=T(I+iB)T^{\ast },$$ 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\u{\i}n 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.