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

DOI10.1007/S44146-022-00027-WzbMATH Open1516.47002arXiv2103.15699WikidataQ114215811 ScholiaQ114215811MaRDI QIDQ6364035

Henk de Snoo, Seppo Hassi

Publication date: 29 March 2021

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 $A$.

Mathematics Subject Classification ID

General (adjoints, conjugates, products, inverses, domains, ranges, etc.) (47A05) Structure theory of linear operators (47A65) Operators on Hilbert spaces (general) (47B02)

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