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Symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk

Radomił Baran, Hugo J. Woerdeman

TL;DR

The work links symmetric Schur-class functions on the bidisk $\\mathbb D^2$ with Schur-class functions on the symmetrized bidisk $\\mathbb G$ by transferring symmetric realizations through $(s,p)=(z+\\zeta, z\\zeta)$. It proves finite-dimensional realizations for symmetric $f$ and, consequently, finite-dimensional realizations for $g$ on $\\mathbb G$ when $f$ is rational; it also develops determinantal representations for symmetric polynomials and a symmetric Nevanlinna-Pick interpolation theory, reducible to bidisk data and solvable via semidefinite programming. The results yield practical, computable models for multivariable Schur-class theory, with applications to interpolation and control-like problems on the symmetrized bidisk. Overall, the paper provides a coherent framework connecting symmetry, realization theory, determinantal representations, and NP interpolation across the bidisk and symmetrized bidisk.

Abstract

We present some thoughts on the relation between symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk. Among other things, use of this relation leads to a finite dimensional realization result for rational matrix functions in the Schur-class on the symmetrized bidisk and also to a determinantal representation result for polynomials without zeros on the symmetrized bidisk.

Symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk

TL;DR

The work links symmetric Schur-class functions on the bidisk with Schur-class functions on the symmetrized bidisk by transferring symmetric realizations through . It proves finite-dimensional realizations for symmetric and, consequently, finite-dimensional realizations for on when is rational; it also develops determinantal representations for symmetric polynomials and a symmetric Nevanlinna-Pick interpolation theory, reducible to bidisk data and solvable via semidefinite programming. The results yield practical, computable models for multivariable Schur-class theory, with applications to interpolation and control-like problems on the symmetrized bidisk. Overall, the paper provides a coherent framework connecting symmetry, realization theory, determinantal representations, and NP interpolation across the bidisk and symmetrized bidisk.

Abstract

We present some thoughts on the relation between symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk. Among other things, use of this relation leads to a finite dimensional realization result for rational matrix functions in the Schur-class on the symmetrized bidisk and also to a determinantal representation result for polynomials without zeros on the symmetrized bidisk.
Paper Structure (3 sections, 6 theorems, 44 equations)

This paper contains 3 sections, 6 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Symmetric functions
  3. The symmetrized bidisk

Key Result

Theorem 2.1

We have that $f(z,\zeta)\in {\mathcal{S}}_{{\mathbb D}^2}({\mathcal{U}}, {\mathcal{Y}})$ is symmetric if and only if there exists a contractive colligation matrix so that Moreover, if $f$ is a rational matrix function then the colligation matrix can be chosen to act on finite dimensional spaces.

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 3 more