Some Representation Theorems for Sesquilinear Forms

TL;DR

This paper investigates representation theorems for sesquilinear forms, establishing conditions under which such forms admit Radon-Nikodym type representations and decompositions, extending classical results to more general non-positive forms.

Contribution

It provides necessary and sufficient conditions for sesquilinear forms to be $ heta$-regular, including a Cauchy-Schwarz inequality criterion, and explores operator representations on Hilbert spaces.

Findings

Characterization of $ heta$-regular sesquilinear forms

Radon-Nikodym type representation criteria

Conditions for operator representation on Hilbert spaces

Abstract

The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$ defined on $\D$, is explored. The main result consists in showing that a sesquilinear form $\Omega$ is $\Theta$-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 $\Theta$-absolutely continuous. In the particular case where $\Theta$ is an inner product in $\mathcal D$, this class of sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace $\mathcal D$ of Hilbert space $\mathcal H$ we give a sufficient condition for the equality $\Omega(\xi,\eta)=\langle{T\xi}|{\eta}\rangle$, with $T$ a closable operator, to hold on a dense subspace of $\mathcal H$.