Functional models, unitary invariants, mosaics and principal functions for operators with trace class self-commutator

The singular integral model of an operator with trace class self-commutator is constructed. A decomposition theorem for operators in orthogonal sum of normal and completely nonnormal operators is obtained. A notion of the hyponormal metrical colligation is introduced, and it is proved that the determining and characteristic functions are unitary invariants for the simple colligations. A class of functions, which are the semidetermining functions is described. On the base of the singular integral model the mosaics and the Pincus principal functions are constructed.