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arXiv · 2603.08559

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

Kelly Bickel·Katie Quertermous·Matina Trachana
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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.

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Kelly Bickel, Katie Quertermous, Matina Trachana. 2026-03-09. Two-Variable Compressions of Shifts, Toeplitz Operators, and Numerical Ranges. https://arxiv.org/abs/2603.08559

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