A Lebesgue-type decomposition for non-positive sesquilinear forms

TL;DR

This paper develops a Lebesgue-type decomposition for non-positive sesquilinear forms, extending classical measure theory concepts to a broader class of forms with applications to complex measures.

Contribution

It introduces a novel decomposition framework for non-positive sesquilinear forms, generalizing Lebesgue decomposition beyond positive measures.

Findings

Decomposition into absolutely continuous and singular parts for non-positive forms

Application of the decomposition to complex measure theory

Extension of classical Lebesgue decomposition concepts

Abstract

A Lebesgue-type decomposition of a (non necessarily non-negative) sesquilinear form with respect to a non-negative one is studied. This decomposition consists of a sum of three parts: two are dominated by an absolutely continuous form and a singular non-negative one, respectively, and the latter is majorized by the product of an absolutely continuous and a singular non-negative forms. The Lebesgue decomposition of a complex measure is given as application.