Complementation and Lebesgue-type decompositions of linear operators and relations

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