A class of sectorial relations and the associated closed forms

Let $T$ be a closed linear relation from a Hilbert space $\mathfrak{H}$ to a Hilbert space $\mathfrak{K}$ and let $B\in \mathbf{B}(\mathfrak{K})$ be selfadjoint. It will be shown that the relation $T^{\ast }(I+iB)T$ is maximal sectorial via a matrix decomposition of $B$ with respect to the orthogonal decomposition $\mathfrak{H}=\mathrm{d}\overline{\mathrm{o}\mathrm{m}}{T}^{\ast }\oplus \mathrm{m}\mathrm{u}\mathrm{l}T$. This leads to an explicit expression of the corresponding closed sectorial form. These results include the case where $\mathrm{m}\mathrm{u}\mathrm{l}T$ is invariant under $B$. The more general description makes it possible to give an expression for the extremal maximal sectorial extensions of the sum of sectorial relations. In particular, one can characterize when the form sum extension is extremal.