Spectral constants for the quantum annulus

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 ${\overline{\mathbb{A}}}_r^n$ 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.

Communications on Pure and Applied Analysis, To appear