Spectral constants for the quantum annulus

TL;DR

This paper investigates spectral constants for the quantum annulus, providing new estimates and alternative proofs for existing bounds, with specific results for commuting operators.

Contribution

It introduces new bounds for spectral constants of the quantum annulus and offers two different proofs for an existing estimate, enhancing understanding of spectral set properties.

Findings

New estimates for spectral constants $K(\

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Abstract

We find several new estimates for the spectral constants $K(\mathbb A_r)$ for which a closed annulus $\overline{\mathbb A}_r$ or closed polyannulus $\overline{\mathbb A}^n_r$ is a $K$-spectral set for operators in the quantum annulus $\mathbb Q \mathbb A_r$. We give two alternative proofs to an existing estimate of spectral constant. The first proof capitalizes a dilation theorem due to McCullough and Pascoe, while the second proof involves a certain variety in the Euclidean biball. For commuting and doubly commuting operators in $\mathbb Q \mathbb A_r$, we find upper and lower bounds for the smallest spectral constants.