Complementation and Lebesgue-type decompositions of linear operators and relations

S. Hassi*, H. S. V. de Snoo

*Corresponding author for this work

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Abstract

In this paper, a new general approach is developed to construct and study Lebesgue-type decompositions of linear operators or relations (Formula presented.) in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue-type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of (Formula presented.). The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue-type decomposition for pairs of quadratic forms. The approach built in this paper uses so-called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.

Original languageEnglish
Article numbere12900
Number of pages32
JournalJournal of the London Mathematical Society
Volume109
Issue number5
DOIs
Publication statusPublished - May-2024

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