# Operators on anti-dual pairs: Lebesgue decomposition of positive operators

## TL;DR

This paper develops a general theory for decomposing positive operators on anti-dual pairs into absolutely continuous and singular parts, generalizing earlier Lebesgue-type decompositions and applying to various mathematical objects.

## Contribution

It introduces a unified framework for Lebesgue decomposition of positive operators on anti-dual pairs, with algebraic, topological characterizations and applications to multiple operator classes.

## Key findings

- Established a general Lebesgue-type decomposition theorem.
- Provided algebraic and topological characterizations of absolute continuity and singularity.
- Demonstrated applications to Hilbert space operators, Hermitian forms, and set functions.

## Abstract

In this paper we introduce and study absolute continuity and singularity of positive operators acting on anti-dual pairs. We establish a general theorem that can be considered as a common generalization of various earlier Lebesgue-type decompositions. Different algebraic and topological characterizations of absolute continuity and singularity are supplied and also a complete description of uniqueness of the decomposition is provided. We apply the developed decomposition theory to some concrete objects including Hilbert space operators, Hermitian forms, representable functionals, and additive set functions.