Solution of the reconstruction-of-the-measure problem for 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 \(\ell ^2({\mathbb {Z}}_+^2)\).