Realization of functions on the symmetrized bidisc

Jim Agler,N.J Young

doi:10.1016/j.jmaa.2017.04.003

Abstract

We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidiscG=def{(z+w,zw):|z|<1,|w|<1}. As an application we prove a Pick-type theorem giving a criterion for the existence of such a function satisfying a finite set of interpolation conditions.

Highlights

The fascination of the symmetrized bidisc G lies in the fact that much of the classical function theory of the disc D and bidisc D2 generalizes in an explicit way to G, but with some surprising twists
We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidisc
The original motivation for the study of G was its connection with the spectral Nevanlinna–Pick problem [3,5], wherefore the emphasis was on analytic maps from the unit disc D into G. In studying such maps one is inevitably drawn into studying maps from G to D; the duality between these two classes of maps is a central feature of the theory of hyperbolic complex spaces in the sense of Kobayashi [18]

Summary

Introduction

The fascination of the symmetrized bidisc G lies in the fact that much of the classical function theory of the disc D and bidisc D2 generalizes in an explicit way to G, but with some surprising twists. The simplest realization formula provides an elegant connection between function theory (the Schur class of the disc) and contractive operators on Hilbert space If so one immediately obtains a realization formula for the general function φ in the Schur–Agler class of G of the form φ(s) = A + BF (s)(1 − DF (s))−1C for some contractive (or unitary) operator colligation ABCD It is significant for this paper that the symmetrized bidisc cannot be defined by a matrix-valued holomorphic function [19], nor is it known to be defined by an operator-valued holomorphic function. We are grateful to an anonymous referee for some very helpful remarks which enabled us to improve the presentation of this paper

A model formula for G

The realization formula

A second model formula for G and spectral domains

A Pick theorem for G