Few maps in the rich structure for the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$
Dinesh Kumar Keshari, Shubhankar Mandal, Avijit Pal
TL;DR
The paper develops a rich-structure framework for domains $\Gamma_{E(3;3;1,1,1)}$ and $\Gamma_{E(3;2;1,2)}$ associated with $\mu_E$-synthesis for $3\times3$ analytic matrix functions on $\mathbb{D}$. It introduces the SE, Upper E, Upper W, and Right S maps and the UW procedure to connect $\mathcal{S}_{1}(\mathbb{C}^3,\mathbb{C}^3)$ with $\mathcal{S}_{3}(\mathbb{C},\mathbb{C})$ and to express kernels on $\mathbb{D}^3$ as rank-1 objects in a reproducing-kernel Hilbert space framework. These mappings yield a mechanism by which interpolation problems for $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$ can be reduced to standard matricial Nevanlinna-Pick problems, providing concrete solvability criteria in the μ-synthesis setting. The approach leverages Agler’s realization theory and lurking isometries to relate matrix-valued Schur classes, analytic kernels, and rank-1 decompositions across multiple domains, with explicit formulas connecting the various objects. This framework thus offers a structured path to analyze and solve interpolation problems in these rich domains with potential applications in robust control and multivariable operator theory.
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 $μ$-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.