A Survey on Solvable Sesquilinear Forms

Abstract

The aim of this paper is to present a unified theory of many Kato type representation theorems in terms of solvable forms on Hilbert spaces. In particular, for some sesquilinear forms $Ω$ on a dense domain $\mathcal{D}$ one looks for an expression $$ Ω(ξ,η)=\langle Tξ, η\rangle, \qquad \forall ξ\in D(T),η\in \mathcal{D}, $$ where $T$ is a densely defined closed operator with domain $D(T)\subseteq \mathcal{D}$. There are two characteristic aspects of solvable forms. Namely, one is that the domain of the form can be turned into a reflexive Banach space need not be a Hilbert space. The second one is the existence of a perturbation with a bounded form which is not necessarily a multiple of the inner product.