Few maps in the rich structure for the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$

TL;DR

This paper explores the rich structure of certain domains related to $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, constructing specific maps and reducing complex interpolation problems to standard Nevanlinna-Pick problems.

Contribution

It introduces new maps within the rich structure of these domains and connects complex interpolation problems to classical Nevanlinna-Pick theory.

Findings

Constructed $SE$, $UW$, $Upper W$, $Upper E$, and $Right S$ maps.

Reduced interpolation problems to standard matricial Nevanlinna-Pick problems.

Established relations between analytic kernels and the domains.

Abstract

The primary goal of a rich structure for some naturally occurring domains $\mathcal X$ is to connect four naturally occurring objects of analysis in the context of $3\times 3$ analytic matrix functions on $\mathbb D$. Combining this rich structure with the classical realisation formula and Hilbert space models in the sense of Agler, one can effectively construct functions in the space $\mathcal O(\mathbb D,\mathcal X)$ of analytic maps from $\mathbb D$ to $\mathcal X$. This allows one to obtain solvability criteria for two cases of the $\mu$-synthesis problem. We describe few maps in the rich structure. We define $SE$ map between $\mathcal S_{1}(\mathbb C^3,\mathbb C^3)$ and $\mathcal S_{3}(\mathbb C,\mathbb C)$ and establish the relation between $\mathcal{S}_{1}(\mathbb C^3,\mathbb C^3)$ and the set of analytic kernels on $\mathbb{D}^{3}$. We obtain the $UW$ procedure and using the $UW$ procedure we construct the $Upper \,\,W$ and $Upper\,\ E$ maps. We also construct $Right~S$ and $SE$ maps. We show how the interpolation problems for $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$ can be reduced to a standard matricial Nevanlinna-Pick problem.