A note on a spectral constant associated with an annulus

TL;DR

This paper investigates the spectral constant associated with an annulus in the complex plane, establishing a lower bound of 2 for this constant across all annuli with radius greater than 1, refining previous bounds.

Contribution

The paper proves that the spectral constant for an annulus is at least 2 for all radii greater than 1, improving upon earlier bounds by Badea, Beckermann, and Crouzeix.

Findings

Established that K(R) ≥ 2 for all R > 1.

Improved previous bounds on spectral constants for annuli.

Refined understanding of spectral set properties for operators.

Abstract

Fix $R>1$ and let $A_R=\{1/R\le |z|\le R \}$ be an annulus. Also, let $K(R)$ denote the smallest constant such that $A_R$ is a $K(R)$-spectral set for the bounded linear operator $T\in \mathcal{B}(H)$ whenever $||T||\le R$ and $||T^{-1}||\le R.$ We show that $K(R)\ge 2, \text{ for all } R>1. $ This improves on previous results by Badea, Beckermann and Crouzeix.