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

2020/01/01 by Raúl E. Curto, Sang Hoon Lee, Curto, Raul E. +3
Engineering · Mathematics · #47 #Advanced Numerical Analysis Techniques #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Approximation and Integration #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2009.06715

openalex publication_date 2020/09/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/06/11

Abstract

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting\n2-variable weighted shifts W_(\α,\β), when the initial data are\ngiven as the Berger measure of the restriction of W_(\α,\β) to a\ncanonical invariant subspace, together with the marginal measures for the 0-th\nrow and 0-th column in the weight diagram for W_(\α,\β). We prove\nthat the natural necessary conditions are indeed sufficient. When the initial\ndata correspond to a soluble problem, we give a concrete formula for the Berger\nmeasure of W_(\α,\β). Our strategy is to build on previous results\nfor back-step extensions and one-step extensions. A key new theorem allows us\nto solve ROMP for two-step extensions. This, in turn, leads to a solution of\nROMP for arbitrary canonical invariant subspaces of \ℓ²(\ℤ₊²).\n

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