Lebesgue type decompositions and Radon-Nikodym derivatives for pairs of bounded linear operators

TL;DR

This paper develops a framework for Lebesgue type decompositions of pairs of bounded linear operators on Hilbert spaces, including a parametrization, criteria for uniqueness, and an abstract Radon-Nikodym derivative.

Contribution

It introduces a comprehensive parametrization of Lebesgue type decompositions for operator pairs and characterizes their uniqueness and Radon-Nikodym derivatives.

Findings

Complete parametrization of all Lebesgue type decompositions.

Characterization of the uniqueness of these decompositions.

Existence of an abstract Radon-Nikodym derivative for the almost dominated part.

Abstract

For a pair of bounded linear Hilbert space operators $A$ and $B$ one considers the Lebesgue type decompositions of $B$ with respect to $A$ into an almost dominated part and a singular part, analogous to the Lebesgue decomposition for a pair of measures (in which case one speaks of an absolutely continuous and a singular part). A complete parametrization of all Lebesgue type decompositions will be given, and the uniqueness of such decompositions will be characterized. In addition, it will be shown that the almost dominated part of $B$ in a Lebesgue type decomposition has an abstract Radon-Nikodym derivative with respect to the operator $A$.