# Representation Theorems for Solvable Sesquilinear Forms

## TL;DR

This paper extends the theory of solvable sesquilinear forms, establishing their structure, properties of associated operators, and criteria for solvability, including relations to numerical range and inner products.

## Contribution

It introduces new results on the structure and criteria of q-closed and solvable sesquilinear forms, generalizing existing theories and analyzing forms related to inner products.

## Key findings

- The structure of the Banach space associated with a q-closed form is unique up to isomorphism.
- The associated operator is the greatest representing the form and is self-adjoint if the form is symmetric.
- New criteria for solvability are provided, including conditions related to the numerical range.

## Abstract

New results are added to the paper [4] about q-closed and solvable sesquilinear forms. The structure of the Banach space $\mathcal{D}[||\cdot||_\Omega]$ defined on the domain $\mathcal{D}$ of a q-closed sesquilinear form $\Omega$ is unique up to isomorphism, and the adjoint of a sesquilinear form has the same property of q-closure or of solvability. The operator associated to a solvable sesquilinear form is the greatest which represents the form and it is self-adjoint if, and only if, the form is symmetric. We give more criteria of solvability for q-closed sesquilinear forms. Some of these criteria are related to the numerical range, and we analyse in particular the forms which are solvable with respect to inner products. The theory of solvable sesquilinear forms generalises those of many known sesquilinear forms in literature.