Spectral constants for the quantum annulus

TL;DR

The paper addresses the problem of determining spectral constants $K$ for which the closed annulus $\overline{\mathbb{A}}_r$ and its multivariable analogues are $K$-spectral sets for operators in the quantum annulus $Q\mathbb{A}_r$. It provides two independent proofs of the Pascoe bound in the one-variable setting—one via a dilation theorem (McCullough–Pascoe) and one via a biball variety—and extends the analysis to multivariable contexts using Ando-type inequalities for commuting and doubly commuting tuples. The authors derive explicit constants for the biannulus ($K=4+\left(\frac{r^2+1}{r^2-1}\right)^2+4\left(\frac{r^2+1}{r^2-1}\right)^{1/2}$) in the commuting-pair case and for the polyannulus ($K_n=\left(\frac{3r^2-1}{r^2-1}\right)^n$) in the doubly commuting $n$-tuple case, along with sharp asymptotic bounds as $r\to\infty$. The work advances understanding of spectral-set phenomena in quantum-annulus settings, providing concrete, computable constants and multiple methodological viewpoints that could inform further dilation and multivariable operator theory research.

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.

Theorems & Definitions (26)

• Proposition 1.1: Pas-McCull, Proposition 3.2
• Theorem 1.2: Pas-McCull, Theorem 1.1 & Proposition 3.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• proof
• Proposition 2.3
• proof
• Corollary 2.4