Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in [formula omitted

Abstract

We consider a bitangential interpolation problem for operator-valued functions defined on a general class of domains in C n (including as particular cases, Cartan domains of types I–III) which satisfy a type of von Neumann inequality associated with the domain. We show that any such function has a realization in terms of a unitary colligation and the defining polynomial for the domain. We show how the solution of various classes of bitangential interpolation problems for this class of functions corresponds to a unitary extension of a particular partially defined isometry uniquely specified by the interpolation data. Criteria for existence of solutions are given (1) in terms of positivity of a certain kernel completely determined by the data, or, more generally, (2) by the existence of a positive-kernel solution of a certain generalized Stein equation completely determined by the data.