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

For a pair of bounded linear Hilbert space operators A and Bone considers the Lebesgue type decompositions of B with respect toA into an almost dominated part and a singular part, analogous to theLebesgue decomposition for a pair of measures in which case one speaks ofan absolutely continuous and a singular part. A complete parametrizationof all Lebesgue type decompositions will be given, and the uniqueness ofsuch decompositions will be characterized. In addition, it will be shownthat the almost dominated part of B in a Lebesgue type decomposition hasan abstract Radon–Nikodym derivative with respect to the operator A.