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 defined on the domain of a q-closed sesquilinear form 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

A Kato's second type representation theorem for solvable sesquilinear forms

“…This extends the representation theorems for sesquilinear forms considered by many authors, as for instance by Kato [11], McIntosh [12], Fleige et al [5,6,7], Grubisić et al [8] and Schmitz [15] (for a more complete list see the references of [2]). For a non-negative closed form Ω, with positive associated operator T , Kato also proved the so-called second representation theorem [11 In the case where Ω is a general sectorial closed form, Kato [10] generalized the representation as…”

“…We conclude this section with a simple relation between a sesquilinear form and its adjoint which gives also another proof of Theorem 4.11 of [2] (in the case of q-closed/solvable forms with respect to an inner product). Then, Ω * has a Radon-Nikodym-like representation…”

“…We recall some definitions and properties concerning q-closed and solvable forms established in [1,2]. …”

“…product with respect to which Ω is q-closed, and following Section 6 of [2], there exists an operator P ∈ B(E Ω ), where…”

“…We call Ω solvable if a perturbation of Ω with a bounded form Υ on H, defines a bounded operator, with bounded inverse, which acts on the triplet (the set of these perturbations is denoted by P(Ω)). These sesquilinear forms have been studied by Di Bella and Trapani in [1] and by Trapani and the author in [2].…”