Geometric Dilations and Operator Annuli

TL;DR

This paper develops a geometric dilation theory for the quantum annulus, characterizing operators with bounded norms and inverses, and compares it with other classical dilation theories in operator analysis.

Contribution

It introduces a geometric approach to dilation for the quantum annulus and compares it with existing dilation theories for other operator annuli.

Findings

Dilation theory for the quantum annulus is fully characterized.

The geometric approach applies broadly to other dilation theorems.

Comparison reveals similarities and differences with classical dilation theories.

Abstract

Fix 1<R. The dilation theory for the quantum annulus, consisting of those invertible Hilbert space operators T such that the norm of T and its inverse are both at most R is determined. The proof technique involves a geometric approach to dilation that applies to other well known dilation theorems. The dilation theory for the quantum annulus is compared, and contrasted, with the dilation theory for other canonical operator annuli.