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Arlinskii's Iteration and its Applications

Part of: General theory of linear operators

Published online by Cambridge University Press:  29 August 2018

Tamás Titkos
Tamás Titkos*
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, Hungary (titkos.tamas@renyi.mta.hu)
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Abstract

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.

Keywords

parallel sumLebesgue decompositionsingularity

MSC classification

Primary: 47A07: Forms (bilinear, sesquilinear, multilinear)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 125 - 133
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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