A new approach to the 2-variable subnormal completion problem

TL;DR

This paper introduces a new method for solving the 2-variable subnormal completion problem using truncated moment problem techniques, establishing sufficiency of conditions and explicitly computing the Berger measure.

Contribution

It develops a general strategy for SCP using moment problem tools and shows sufficiency of necessary conditions when quadratic moments are known.

Findings

Necessary conditions for subnormal completion are sufficient in the quadratic moments case.

Explicit computation of the Berger measure via algebraic variety analysis.

Provides a new algebraic approach to the 2-variable SCP.

Abstract

We study the Subnormal Completion Problem (SCP) for 2-variable weighted shifts. We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion.