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

TL;DR

This paper explores two-variable shift compressions linked to rational inner functions on the bidisk, revealing their relation to matrix-valued Toeplitz operators and examining the properties of their numerical ranges.

Contribution

It introduces a new class of two-variable compressions, establishes their unitary equivalence to matrix-valued Toeplitz operators, and analyzes the extent to which these functions are determined by their numerical ranges.

Findings

Rational inner functions are almost determined by their Toeplitz symbols.

Rational inner functions are not fully determined by the numerical ranges of their compressed shifts.

Conditions for the openness and closedness of numerical ranges are characterized.

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.