Models of holomorphic functions on the symmetrized skew bidisc

TL;DR

This paper develops a theoretical framework for bounded holomorphic functions on the symmetrized skew bidisc, including realization and model formulas, expanding understanding of these functions in complex analysis.

Contribution

It introduces new realization and model formulas for holomorphic functions on the symmetrized skew bidisc, a significant extension in complex function theory.

Findings

Established the existence of a realization formula.

Derived a model formula for holomorphic functions.

Enhanced the theoretical understanding of functions on the symmetrized skew bidisc.

Abstract

The purpose of this paper is to develop the theory of holomorphic functions with modulus bounded by $1$ on the symmetrized skew bidisc \[ \mathbb{G}_{r} \stackrel{\rm def}{=} \Big\{( \lambda_{1}+r\lambda_{2} ,r\lambda_{1}\lambda_{2}): \lambda_{1}\in \mathbb{D}, \lambda_{2}\in\mathbb{D}\Big\}, \] for a fixed $r \in (0,1)$. We show the existence of a realization formula and a model formula for such holomorphic functions.