Operators on anti-dual pairs: Lebesgue decomposition via Arlinskii's iteration

TL;DR

This paper develops a general Lebesgue decomposition theorem for positive operators on anti-dual pairs using Arlinskii's iterative method, unifying various special cases like Hilbert space operators and *-algebra functionals.

Contribution

It introduces a unified approach to Lebesgue decomposition for positive operators on anti-dual pairs, extending previous results to broader contexts.

Findings

Proves a general Lebesgue decomposition theorem for positive operators.

Demonstrates the theorem encompasses Hilbert space operators and *-algebra functionals.

Utilizes Arlinskii's iterative approach for the decomposition process.

Abstract

The aim of this paper is to prove a general Lebesgue decomposition theorem for positive operators on so-called anti-dual pairs, following the iterative approach introduced by Arlinskii. This procedure and the resulting theorem encompass several special cases, including positive operators on Hilbert spaces, non-negative forms on vector spaces, and representable functionals over *-algebras.