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,⟨.,.⟩) In particular, for some sesquilinear forms Ω on a dense domain D ⊆ H one looks for a representation Ω(ξ, η) = ⟨Tξ, η⟩ (ξ ϵ D(T), η ϵ D), where T is a densely defined closed operator with domain D(T) ⊆ 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 reflexive 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.