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On interpolation in Carathéodory hyperbolic domains

Anindya Biswas

TL;DR

The paper addresses the problem of describing Pick bodies on Carathéodory hyperbolic domains and realizing them via unit disc models. It develops a Schur–Agler–type framework showing $\mathscr{D}_\Omega(\underline{z})$ equals the intersection $\bigcap_{K\in\mathcal{K}_\Omega}\mathscr{D}_K$ over admissible positive definite kernels and establishes a 3-point sufficiency criterion for disc-model realization using a boundary point $\underline{\alpha}$ and the generalized Carathéodory function $c_\Omega^*$. It also analyzes extremal kernels, $K(z_i,z_i)=1$, and the associated rank/phase structure to deduce conditions under which $\mathscr{D}_\Omega(\underline{z})=\mathscr{D}_\mathbb{D}(\underline{\alpha})$. These results extend the understanding of Pick interpolation from the bidisc to general Carathéodory hyperbolic domains and provide practical criteria for disc-model realizations.

Abstract

We study the relation between Pick bodies on Carathéodory hyperbolic domains and contractions on finite dimensional Hilbert spaces. We give a condition sufficient to realize Pick bodies on Carathéodory hyperbolic domains as a Pick body on the open unit disc.

On interpolation in Carathéodory hyperbolic domains

TL;DR

The paper addresses the problem of describing Pick bodies on Carathéodory hyperbolic domains and realizing them via unit disc models. It develops a Schur–Agler–type framework showing equals the intersection over admissible positive definite kernels and establishes a 3-point sufficiency criterion for disc-model realization using a boundary point and the generalized Carathéodory function . It also analyzes extremal kernels, , and the associated rank/phase structure to deduce conditions under which . These results extend the understanding of Pick interpolation from the bidisc to general Carathéodory hyperbolic domains and provide practical criteria for disc-model realizations.

Abstract

We study the relation between Pick bodies on Carathéodory hyperbolic domains and contractions on finite dimensional Hilbert spaces. We give a condition sufficient to realize Pick bodies on Carathéodory hyperbolic domains as a Pick body on the open unit disc.
Paper Structure (2 sections, 8 theorems, 29 equations)

This paper contains 2 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction and statements of the results
  2. Proofs of the results

Key Result

Theorem 1.1

For any Carathéodory hyperbolic domain $\Omega$ and $n$ distinct points $z_1,\ldots, z_n$ in $\Omega$, there is a collection $\mathcal{K}_\Omega$ of positive definite $n\times n$ matrices such that

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Definition
  • Definition
  • Definition
  • Example
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['PickBodyAsIntersection']}
  • Proposition 2.1
  • ...and 10 more