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 a Hilbert space $(H,\langle\cdot,\cdot\rangle)$ In particular, for some sesquilinear forms Ω on a dense domain $D\subseteq\mathcal {H}$ one looks for a representation $\Omega(\xi,\eta)= \langle T\xi,\eta\rangle$ $(\xi\epsilon\mathcal{D}\mathcal(T),\eta\epsilon D)$ where T is a densely defined closed operator with domain $D(\mathcal{T})\subseteq \mathcal{D}$. There are two characteristic aspects of a solvable form on H. One is that the domain of the form can be turned into a reexive Banach space that need not be a Hilbert space. The second one is that representation theorems hold after perturbing the form by a bounded form that is not necessarily a multiple of the inner product of H.