Complementation and Lebesgue type decompositions of linear operators and relations

TL;DR

This paper introduces a novel, more general method for Lebesgue type decompositions of linear operators in Hilbert spaces, enabling richer interactions between components and connecting to quadratic form decompositions.

Contribution

It develops a new approach using complementation in Hilbert spaces, broadening the class of Lebesgue type decompositions for operators and relations.

Findings

Allows wider class of decompositions than previous methods

Enables nontrivial interaction between closable and singular parts

Links operator decompositions to quadratic form decompositions

Abstract

In this paper a new general approach is developed to construct and study Lebesgue type decompositions of linear operators $T$ 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 $T$. 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.