Models of holomorphic functions on the symmetrized skew bidisc
Connor Evans, Zinaida A. Lykova, N. J. Young
TL;DR
Problem: study holomorphic functions with modulus <= 1 on the symmetrized skew bidisc $G_r$. Approach: develop model and realization formulas by transferring the disc/bidisc framework to $G_r$ via the maps $T_r$ and $\pi$ and a symmetry $\sigma$, yielding operator-valued kernels built from $s_{U,R}$. Contributions: existence of $G_r$-models and $G_r$-realizations for every $f \in \mathcal S(G_r)$, explicit holomorphic dependence on the parameter $r$, and concrete finite-dimensional examples such as $\Upsilon_{\omega,r}$ and $\Phi_{\omega}$ illustrating the theory. Significance: provides interpolation and control-theoretic tools on non-convex domains, extending Agler–Young techniques to $G_r$ and linked to spectral Nevanlinna-Pick problems.
Abstract
The purpose of this paper is to develop the theory of holomorphic functions with modulus bounded by $1$ on the symmetrized skew bidisc \[ \mathbb{G}_{r} \stackrel{\rm def}{=} \Big\{( λ_{1}+rλ_{2} ,rλ_{1}λ_{2}): λ_{1}\in \mathbb{D}, λ_{2}\in\mathbb{D}\Big\}, \] for a fixed $r \in (0,1)$. We show the existence of a realization formula and a model formula for such holomorphic functions.