Arlinskii's iteration and its applications

Abstract

Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called: parallel sum. The aim of this paper is to investigate this relation from a new point of view. Namely, using a natural generalization of Arlinskii's approach (which identifies the singular part as a fixed point of a single-variable map) we prove the existence of a Lebesgue-type decomposition for nonnegative sesquilinear forms. As applications, we also show that how this approach can be used to derive analogous results for representable functionals, nonnegative finitely additive measures, and positive definite operator functions. The focus is on the fact that each theorem can be proved with the same completely elementary method.