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

Authors

Raul E. Curto
Sang H. Lee
Jasang Yoon, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

6-2022

Abstract

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts W(α,β), when the initial data are given as the Berger measure of the restriction of W(α,β) 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(α,β). 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(α,β). Our strategy is to build on previous results for back-step extensions and onestep 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 ℓ2(Z2 +).

Recommended Citation

Curto, R.E., Lee, S.H. & Yoon, J. Solution of the reconstruction-of-the-measure problem for canonical invariant subspaces. Annali di Matematica 201, 1489–1504 (2022). https://doi.org/10.1007/s10231-021-01166-7

Publication Title

Annali di Matematica

DOI

https://doi.org/10.1007/s10231-021-01166-7