  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 $A$.
