Recursive relations for 2-variable weighted shifts

TL;DR

This paper presents a method to derive recursive relations for 2-variable weighted shifts from their representing measure, enabling explicit construction of a Gr"obner basis for these relations.

Contribution

It introduces a novel approach to determine recursive relations from the representing measure and constructs a Gr"obner basis for the ideal of these relations.

Findings

Recursive relations depend only on the representing measure, not atom densities.

Method allows explicit construction of Gr"obner basis for recursive relations.

Applicable to subnormal 2-variable weighted shifts.

Abstract

In this paper, we outline a method to determine all recursive relations for a subnormal 2-variable weighted shift, up to total degree $k$, entirely from the representing measure. This allows us to show that the densities of the atoms do not affect the recursive relations. We then show that this method can be used to explicitly construct a Gr\"obner basis for the ideal of recursive relations.