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

Abstract

There are three new things in this paper about the open symmetrized bidisk . 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 . (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 . 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 -norm preserving extension to the whole of .