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In this paper a new general approach is developed to construct and studyLebesgue type decompositions of linear operators $T$ in the Hilbert spacesetting. The new approach allows to introduce an essentially wider class ofLebesgue type decompositions than what has been studied in the literature sofar. The key point is that it allows a nontrivial interaction between theclosable and the singular components of $T$. The motivation to study suchdecompositions comes from the fact that they naturally occur in thecorresponding Lebesgue type decomposition for pairs of quadratic forms. Theapproach built in this paper uses so-called complementation in Hilbert spaces,a notion going back to de Branges and Rovnyak.

Complementation and Lebesgue type decompositions of linear operators and relations