Solution of the reconstruction-of-the-measure problem for canonical invariant subspaces

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 $\ell^2(\mathbb{Z}_+^2)$.