Holomorphic functions on the symmetrized bidisk – Realization, interpolation and extension

Abstract

There are three new things in this paper about the open symmetrized bidisk G={(z1+z2,z1z2):|z1|,|z2|<1}. They are, in the order in which they will be proved,(1)The Realization Theorem: A realization formula is demonstrated for every f in the norm unit ball of H∞(G).(2)The Interpolation Theorem: A Nevanlinna–Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function.(3)The Extension Theorem: Let V be a subset of the symmetrized bidisk G. Consider a function f that is holomorphic in a neighbourhood of V and bounded on V. A necessary and sufficient condition on f is obtained so that f possesses an H∞-norm preserving extension to the whole of G.