Some representation theorems for sesquilinear forms

Abstract

The possibility of getting a Radon–Nikodym type theorem and a Lebesgue-like decomposition for a not necessarily positive sesquilinear Ω form defined on a vector space D, with respect to a given positive form Θ defined on D, is explored. The main result consists in showing that a sesquilinear form Ω is Θ-regular, in the sense that it has a Radon–Nikodym type representation, if and only if it satisfies a sort Cauchy–Schwarz inequality whose right hand side is implemented by a positive sesquilinear form which is Θ-absolutely continuous. In the particular case where Θ is an inner product in D, this class of sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace D of Hilbert space H we give a sufficient condition for the equality Ω(ξ,η)=〈Tξ|η〉, with T a closable operator, to hold on a dense subspace of H

Biorthogonal vectors, sesquilinear forms, and some physical operators

“…Thus, Kato's representation theorems cannot be applied. This would be not a major problem since several variants to these famous theorems have been proposed (we refer to [22,23], where the notion of solvable form has been introduced and studied, and for a rather complete bibliography on this matter). However, all this is of little use for us since (3.4) implies that Ω ϕ,ψ and Ω ψ,ϕ are both positive.…”

Arlinskii's Iteration and its Applications

“…To see that this type of decomposition is not unique in general we refer the reader to [15,Theorem 4.4]. An analogous decomposition of not necessarily nonnegative sesquilinear forms can be found in the recent paper of Di Bella and Trapani [8,Section 4].…”

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

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

“…As proved in [1], for a solvable sesquilinear form Ω there exists a closed operator T , with dense domain D(T ) ⊆ D, such that the following representation holds Ω(ξ, η) = T ξ, η , ∀ξ ∈ D(T ), η ∈ D.…”

“…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].…”