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arXiv · 2009.06715

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

Raul E. Curto·Sang Hoon Lee·Jasang Yoon
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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 $\ell^2(\mathbb{Z}_+^2)$.

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Raul E. Curto, Sang Hoon Lee, Jasang Yoon. 2020-09-14. Solution of the Reconstruction-of-the-Measure Problem for Canonical Invariant Subspaces. https://doi.org/10.1007/s10231-021-01166-7

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