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.
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Hassi, S., De Snoo, H. Lebesgue type decompositions and Radon–Nikodym derivatives for pairs of bounded linear operators. Acta Sci. Math. (Szeged) 88, 469–503 (2022). https://doi.org/10.1007/s44146-022-00027-w
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DOI: https://doi.org/10.1007/s44146-022-00027-w
Key words and phrases
- Lebesgue type decompositions
- operator range
- pair of bounded operators
- almost dominated part
- singular part
- Radon–Nikodym derivative