Dilation on an annulus and von Neumann's inequality on certain varieties in the biball

TL;DR

This paper provides an alternative proof of rational dilation success on an annulus, exploring operator theory on varieties in the bidisk and their spectral properties, connecting classical and quantum annulus operators.

Contribution

It introduces a new approach to rational dilation on an annulus using operator pairs on varieties in the bidisk, linking classical and quantum operator models.

Findings

Alternative proof of rational dilation on an annulus.

Operator interplay between classical and quantum annulus.

Spectral set and von Neumann's inequality results for these classes.

Abstract

We give an alternative proof to Agler's famous result on success of rational dilation on an annulus by an application of a result due to Dritschel and McCullough. We show interplay between operators associated with an annulus, $C_{1,r}$ or quantum annulus and operator pairs living on a certain variety in $\mathbb C^2$ and its intersection with the biball. It is shown that the minimal spectral sets and von Neumann's inequality for these classes $C_{1,r}$, quantum annulus can also be studied via appropriate operator pairs associated with the biball.