Compressions of the Shift on the Bidisk and their Numerical Ranges

TL;DR

This paper studies the properties of compressed shift operators on bidisk model spaces, characterizing their numerical ranges and radii, especially for simple rational inner functions, revealing geometric and spectral insights.

Contribution

It introduces a unitary equivalence between compressed shifts and matrix-valued Toeplitz operators, providing new tools to analyze their numerical ranges and geometric properties.

Findings

Compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators.

Explicit formulas for numerical range boundaries are derived.

Conditions for circular numerical ranges are established.

Abstract

We consider two-variable model spaces associated to rational inner functions $\Theta$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_{\Theta}:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and study the geometry of the associated numerical ranges, find formulas for the boundaries, answer the zero inclusion question, and determine whether the numerical ranges are ever circular.