A New Proof that the Numerical Range is a Complete 2-Spectral Set for Weighted Shift Matrices

TL;DR

This paper provides an alternative proof that certain weighted shift matrices satisfy Crouzeix's conjecture and its completely bounded version, advancing understanding of spectral set properties.

Contribution

It offers a new proof that the numerical range is a complete 2-spectral set for these matrices, building on prior work by Choi.

Findings

Matrices satisfy Crouzeix's conjecture

Matrices satisfy the completely bounded version of the conjecture

Provides an alternative proof for spectral set properties

Abstract

In this paper we give an alternative proof that the family of matrices studied by Daeshik Choi in A proof of Crouzeix's conjecture for a class of matrices, Linear Algebra and its Applications, 438, no. 8 (2013), pp. 3247-3257, satisfy Crouzeix's conjecture. We also show that they satisfy the completely bounded version of the conjecture.