A Class of Sectorial Relations and the Associated Closed Forms

Abstract

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 ^∗(I + iB)T is maximal sectorial via a matrix decomposition of B with respect to the orthogonal decomposition ${\mathfrak H}=\mathrm {d}\overline {\mathrm {om}}\, T^* \oplus \mathrm {mul}\, T$. This leads to an explicit expression of the corresponding closed sectorial form. These results include the case where mul 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.

Dedicated to V.E. Katsnelson on the occasion of his 75th birthday

The second author thanks the University of Vaasa for its hospitality during the preparation of this work.