Operators on anti-dual pairs: Lebesgue decomposition of positive operators

Abstract

In this paper we introduce and study absolute continuity and singularity of positive operators acting on anti-dual pairs. We establish a general theorem that can be considered as a common generalization of various earlier Lebesgue-type decompositions. Different algebraic and topological characterizations of absolute continuity and singularity are supplied and also a complete description of uniqueness of the decomposition is provided. We apply the developed decomposition theory to some concrete objects including operators acting in a rigged Hilbert space, Hermitian forms, representable functionals, and additive set functions.