Numerical Range and Compressions of the Shift

TL;DR

This survey explores the numerical ranges of operators, especially the shift operator and its compressions, highlighting their properties, connections to dilations, and related conjectures in operator theory.

Contribution

It provides a comprehensive overview of numerical ranges of shift operators, their compressions, and related results, including connections to unitary dilations and the Crouzeix conjecture.

Findings

Numerical ranges of matrices relate to envelopes of curve families.

Connections between numerical ranges of shift operators and their dilations are established.

Results on compressed shift operators on the bidisk are discussed.

Abstract

The numerical range of a bounded, linear operator on a Hilbert space is a set in $\mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(\mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 \ominus \theta H^2$ where $\theta$ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture.