Weighted Schur-class functions and functional models

Abstract

We generalize the class of contractive-valued functions analytic in the unit disk (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 the orthogonal sum of the pure and unitary parts) are extended to our setting. This generalization allows one to use a uniform approach to functional models developed recently in papers of the author and D. Yakubovich. Moreover, it gives an alternative language to handle multi-valued functions in the multiply connected case.