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.1991 Mathematics Subject Classification. Primary 47A07, Secondary 46L51, 28A12, 43A35.

Operators on anti-dual pairs: Generalized Schur complement

“…Accordingly, it is obvious that A f : A g = A f :g and A f ÷ A g = A f ÷g due to (3.16) and (3.17). This simple observation leads us to the following Lebesgue type decomposition theorem (for earlier versions of the Lebesgue decomposition of representable functionals see [7,11,16,18,19,21], and the references therein). Theorem 3.10.…”

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

“…We remark that the existence of a Lebesgue-type decomposition can be proved easily by means of (ii) with an elementary iteration involving parallel addition (see [5] and [45]). Hovewer, the itaration itself does not guarantee the maximality of the resulted absolutely continuous part.…”