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Contractive realization theory for the annulus and other intersections of discs on the Riemann sphere

Radomił Baran, Piotr Pikul, Hugo J. Woerdeman, Michał Wojtylak

TL;DR

This work develops a finite-dimensional contractive realization theory for rational matrix functions on non simply connected domains, notably the annulus, by combining multivariable realization results with operator-algebra techniques. It defines and analyzes the Agler norm on bounded intersections of discs on the Riemann sphere, proves a key norm identity via a polydisk lifting, and establishes a central contractive realization theorem: if $F$ has $\|F\|_{\mathcal{T}_Ω}<1$, then there exists a contractive colligation $ABCD$ giving $F$ via $F(z)=D+C\mathbf{P}_-(z)_{m}(\mathbf{P}_+(z)_{m}-A\mathbf{P}_-(z)_{m})^{-1}B$. The paper then extends to open-domain norms, derives convex and annulus-specific corollaries, and applies the framework to Bohr-type inequalities, yielding new bounds for the Bohr radius on the annulus and bidisk. These results advance interpolation and spectral-set theory on multiply connected domains and connect to practical bounds in Banach-algebra settings.

Abstract

We develop contractive finite dimensional realizations for rational matrix functions of one variable on domains that are not simply connected, such as the annulus. The proof uses multivariable contractive realization results as well as abstract operator algebra techniques. Other results include new bounds for the Bohr radius of the bidisk and the annulus.

Contractive realization theory for the annulus and other intersections of discs on the Riemann sphere

TL;DR

This work develops a finite-dimensional contractive realization theory for rational matrix functions on non simply connected domains, notably the annulus, by combining multivariable realization results with operator-algebra techniques. It defines and analyzes the Agler norm on bounded intersections of discs on the Riemann sphere, proves a key norm identity via a polydisk lifting, and establishes a central contractive realization theorem: if has , then there exists a contractive colligation giving via . The paper then extends to open-domain norms, derives convex and annulus-specific corollaries, and applies the framework to Bohr-type inequalities, yielding new bounds for the Bohr radius on the annulus and bidisk. These results advance interpolation and spectral-set theory on multiply connected domains and connect to practical bounds in Banach-algebra settings.

Abstract

We develop contractive finite dimensional realizations for rational matrix functions of one variable on domains that are not simply connected, such as the annulus. The proof uses multivariable contractive realization results as well as abstract operator algebra techniques. Other results include new bounds for the Bohr radius of the bidisk and the annulus.

Paper Structure

This paper contains 8 sections, 18 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Agler norm on intersection of discs on the Riemann sphere
  3. Contractive realization on intersection of discs on the Riemann sphere
  4. Agler norms with respect to open domains
  5. Special cases
  6. The convex case
  7. The case of the annulus
  8. The Bohr inequality for the annulus and the bidisc

Key Result

Proposition 2.2

Let $\Omega$ be the bounded intersection of discs on the Riemann sphere given by Omegadef. For each $T$ in the class $\mathcal{T}_\Omega$ the set $\Omega$ is a $\Psi_{cb}$-complete spectral set with $\Psi_{cb}\leq k+k(k-1)/\sqrt 3$. In particular, the Agler norm is finite and bounded from above as

Figures (2)

  • Figure 1: Examples of a set $\Omega$ and the corresponding $\check\Omega$ for $k=3,4$.
  • Figure 2: Examples of "multi-holed discs". Note that we do not require the holes to be disjoint.

Theorems & Definitions (37)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 27 more