Representation Theorems for Solvable Sesquilinear Forms

Abstract

New results are added to the paper (Di Bella and Trapani in J Math Anal Appl 451:64–83, 2017) 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.