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The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts
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The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts

Raul E Curto, Youssef Omari, Hamza El Azhar and El Hassan Zerouali
arXiv.org
Cornell University
10/21/2023
DOI: 10.48550/arxiv.2310.13887
url
https://doi.org/10.48550/arxiv.2310.13887View
Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ({\bf SP}) (related to the Aluthge transform) and the Square Root Problem ({\bf SRP}) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For $p$--atomic measures with $p \le 6$, our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a $7$--atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by G.R. Exner in 2009, and to a recent conjecture formulated by R.E. Curto et al in 2019.
 Mathematics - Functional Analysis

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