A rich structure related to the construction of analytic matrix functions

TL;DR

This paper explores the structure of analytic matrix functions related to specific $ ext{-}synthesis$ problems, focusing on interpolation in complex domains like the symmetrised bidisc and tetrablock, using advanced function space theories.

Contribution

It provides a new framework linking $ ext{-}synthesis$ problems with interpolation in specialized complex domains through Hilbert space and kernel methods.

Findings

Established connections between analytic functions, Schur classes, and positive kernels.

Provided a solvability criterion for interpolation problems in the tetrablock domain.

Analyzed the structure of interconnections in complex analytic function spaces.

Abstract

We analyse two special cases of $\mu$-synthesis problems which can be reduced to interpolation problems in the set of analytic functions from the disc into the symmetrised bidisc and into the tetrablock. For these inhomogeneous domains we study the structure of interconnections between the set of analytic functions from the disc into the given domain, the matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. We use the theories of Hilbert function spaces and of reproducing kernels to establish these connections. We give a solvability criterion for the interpolation problem that arises from the $\mu$-synthesis problem related to the tetrablock.