Numerical ranges of $C_0(N)$ contractions

TL;DR

This paper investigates the numerical ranges of $C_0(N)$ contractions, showing that for these operators, the intersection of numerical ranges of a restricted family of dilations equals the closure of the original contraction's numerical range, extending finite-dimensional results.

Contribution

It extends a finite-dimensional result to $C_0(N)$ contractions, demonstrating that a smaller family of dilations suffices to determine the numerical range closure.

Findings

For $C_0(N)$ contractions, the intersection of a smaller dilation family matches the numerical range closure.

Generalizes finite-dimensional results to infinite-dimensional $C_0(N)$ contractions.

Provides a refined understanding of numerical ranges for a class of contractions.

Abstract

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for $C_0(N)$ contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.