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Conservative curved systems over multiply connected domains are introduced and relationships of such systems with related notions (functional model, characteristic function, and transfer function) are studied. In contrast to standard theory for the unit disk, characteristic functions and transfer functions are essentially different objects. We study possibility to recover the characteristic function for a given transfer function. As the result we obtain the procedure to construct the functional model for a given conservative curved system.

We study operator-valued functions of weighted Schur classes over multiply-connected domains. There is a correspondence between functions of weighted Schur classes and so-called “conservative curved” systems introduced in the paper. In the unit disk case the fundamental relationship between invariant subspaces of the main operator of a conservative system and factorizations of the corresponding Schur class function (characteristic function) is well known. We extend this connection to weighted Schur classes. With this aim we develop new notions and constructions and make suitable changes in standard theory.

The aim of this paper is to extend the S.-Nagy-Foias functional model to the case of functions for contractive operators. This model meets general requirements and can be used for studying trace class perturbations of normal operators with spectrum on a curve.

The aim of this paper is to study relations between “curved” conservative systems (systems for which the main operator is a function of contraction) and their transfer functions. We derive the boundedness property for such transfer functions and apply it to some problems of spectral analysis.

A new approach to obtain the existence and uniqueness theorem of inner-outer factorization is proposed. This approach does not employ the Sz.-Nagy-Foias-Naboko functional model nor the Shmul’jan-Pekarev regularity factorization criterion.