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parallel sum

Proceedings of the Edinburgh Mathematical Society (1)

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1 results

Arlinskii's Iteration and its Applications
Part of
- General theory of linear operators
Tamás Titkos
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 1 / February 2019

Published online by Cambridge University Press:

29 August 2018, pp. 125-133
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Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called the 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 non-negative sesquilinear forms. As applications, we also show how this approach can be used to derive analogous results for representable functionals, non-negative 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.