Solution of the Reconstruction-of-the-Measure Problem for Canonical Invariant Subspaces

TL;DR

This paper solves the Reconstruction-of-the-Measure Problem for certain 2-variable weighted shifts by establishing sufficiency of necessary conditions and providing explicit formulas for Berger measures, extending previous extension results.

Contribution

It introduces a comprehensive solution to ROMP for canonical invariant subspaces, including a new theorem for two-step extensions and explicit Berger measure formulas.

Findings

Necessary conditions are sufficient for ROMP.

Explicit Berger measure formulas are derived for solvable cases.

The approach extends to arbitrary canonical invariant subspaces.

Abstract

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts $W_{(\alpha,\beta)}$, when the initial data are given as the Berger measure of the restriction of $W_{(\alpha,\beta)}$ to a canonical invariant subspace, together with the marginal measures for the 0-th row and 0-th column in the weight diagram for $W_{(\alpha,\beta)}$. We prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, we give a concrete formula for the Berger measure of $W_{(\alpha,\beta)}$. Our strategy is to build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspaces of $\ell^2(\mathbb{Z}_+^2)$.