The spectral constant for the quantum cross and asymptotically sharp bounds for annuli

TL;DR

This paper investigates the spectral constant for quantum annuli and crosses, establishing asymptotically sharp bounds and revealing the limit of the spectral constant as the annulus parameter grows.

Contribution

It introduces a new approach to analyze spectral constants in quantum annuli and crosses, providing exact limits and bounds through rational dilation techniques.

Findings

Spectral constant limit as r approaches infinity is 2.

Spectral constant for the quantum cross is exactly 2.

Established bounds for spectral constants in quantum annuli.

Abstract

The quantum annulus of type $r$ is the class of invertible operators with singular values in $(1/r,r).$ Given an analytic function on the classical annulus of type $r,$ we may evaluate it on operators in the quantum annulus by The spectral constant gives the maximum ratio betweeen the supremum over the norm of evalutions at operators in the quantum annulus to the supremum over classical evaluations. We show that the limit of the spectral constant as $r$ goes to infinity is $2.$ Via the correspondence between annuli and hyperbolae, our study degenerates the problem to one on the quantum cross, pairs of contractions with product zero, where the spectral constant is exactly $2.$ The essential technique is to rationally dilate $Z$ to $\hat{Z}$ which has $U =(\hat{Z}+(\hat{Z}^{-1})^*)/(r+1/r)$ unitary and estimate $Uf(\hat{Z})U^*$ directly.