Representing maps for semibounded forms and their Lebesgue-type decompositions

Abstract

In the Lebesgue decomposition of a lower semibounded sesquilinear form, the corresponding regular and singular parts are mutually singular. The more general Lebesgue-type decompositions studied here allow components that need not be mutually singular anymore. In the new situation, the earlier basic orthogonal space decomposition in the background is now replaced by a nonorthogonal decomposition in the sense of de Branges and Rovnyak. The relevant theory is based on Lebesgue-type decompositions for linear operators and relations via a so-called representing map. This map also makes it possible to formulate explicit analogs for representation theorems for lower semibounded forms that are not necessarily closed or closable. This new representation also appears naturally in the convergence of monotone sequences of lower semibounded forms.