Schur Functions and Inner Functions on the Bidisc

Abstract

We study representations of inner functions on the bidisc from a fractional linear transformation point of view. We provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit a one variable factorization. We present a classification of de Branges–Rovnyak kernels on the bidisc (which also works in the setting of the polydisc and the open unit ball of \({\mathbb {C}}^n\), \(n \ge 1\)). We also classify, in terms of Agler kernels, two-variable Schur functions that admit a one variable factorization.