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

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.© 2022 The Authors. Published by Springer. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit (http://creativecommons.org/licenses/by/4.0/).fi=vertaisarvioitu|en=peerReviewed