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Two-Variable Compressions of Shifts, Toeplitz Operators, and Numerical Ranges

Kelly Bickel, Katie Quertermous, Matina Trachana

Abstract

This paper studies two-variable compressions of shifts associated to rational inner functions on the bidisk; these generalize the classical compressions of the shift associated to finite Blasckhe products and are unitarily equivalent to one-variable, matrix-valued Toeplitz operators. This paper proves that a rational inner function is almost completely determined by these Toeplitz operator symbols but provides examples showing that (unlike in the one-variable case) rational inner functions are not determined by the numerical ranges of their compressed shifts. This paper also investigates related questions including methods of constructing these compressed-shift Toeplitz operators and when the associated numerical ranges are open and closed.

Two-Variable Compressions of Shifts, Toeplitz Operators, and Numerical Ranges

Abstract

This paper studies two-variable compressions of shifts associated to rational inner functions on the bidisk; these generalize the classical compressions of the shift associated to finite Blasckhe products and are unitarily equivalent to one-variable, matrix-valued Toeplitz operators. This paper proves that a rational inner function is almost completely determined by these Toeplitz operator symbols but provides examples showing that (unlike in the one-variable case) rational inner functions are not determined by the numerical ranges of their compressed shifts. This paper also investigates related questions including methods of constructing these compressed-shift Toeplitz operators and when the associated numerical ranges are open and closed.
Paper Structure (10 sections, 19 theorems, 153 equations, 1 figure)

This paper contains 10 sections, 19 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. One-variable setting
  3. Two-variable setting
  4. Overview of Paper and Key Results
  5. Two-variable Details
  6. The Degree $(1,n)$ Case
  7. $M^1_{\Theta}$ for Products of Rational Inner Functions
  8. Uniqueness of $M^1_\theta$
  9. Non-Uniqueness of $W(S_\theta^1)$
  10. Open and closed numerical ranges

Key Result

Theorem 1.1

Let $B_1$, $B_2$ be finite Blaschke products. Then the following are equivalent:

Figures (1)

  • Figure 3.1: The closures of $M^1_\theta(\mathbb{D})$ and $M^1_\phi(\mathbb{D})$ in $\overline{\mathbb{D}}$ for $\theta$ and $\phi$ from \ref{['eqn:exphi']}.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary
  • Theorem
  • Theorem 2.1
  • Example 2.2
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 33 more