Weighted Schur class functions and functional model

We generalize the class of contractive valued, analytic in the unit disk functions (Schur class) to the case of finitely connected domains with operator valued weights on the boundary. The two fundamental decompositions (the inner-outer factorization and the decomposition into an orthogonal sum of the pure and the unitary parts) are extended to our setting. This generalization allows to use an uniform approach to functional models developed recently in papers of the author and D. Yakubovich. Moreover, it gives an alternative language to handle multiply-valued functions in the multiply-connected case.