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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Acknowledgements
The authors are very grateful to the referee for a careful reading of the manuscript and for several suggestions that helped improve the presentation.
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The second author of this paper was partially supported by NRF (Korea) grant No. 2020R1A2C1A0100584611. The third named author was partially supported by a grant from the University of Texas System and the Consejo Nacional de Ciencia y Tecnología de México (CONACYT).
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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
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DOI: https://doi.org/10.1007/s10231-021-01166-7
Keywords
- Two-step extension
- 2-Variable weighted shifts
- Subnormal pair
- Berger measure
- Canonical invariant subspace