A Survey on Solvable Sesquilinear Forms

TL;DR

This survey presents a unified framework for solvable sesquilinear forms on Hilbert spaces, extending Kato's representation theorems and exploring their properties and perturbations.

Contribution

It provides a comprehensive overview of solvable forms, unifying various Kato type theorems and analyzing their domain and perturbation characteristics.

Findings

Unified theory of solvable forms on Hilbert spaces

Characterization of forms with reflexive Banach space domains

Existence of perturbations with bounded 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 $\Omega$ on a dense domain $\mathcal{D}$ one looks for an expression $$ \Omega(\xi,\eta)=\langle T\xi , \eta\rangle, \qquad \forall \xi\in D(T),\eta \in \mathcal{D}, $$ 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.