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Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 241–255 | Cite as

Moment Infinitely Divisible Weighted Shifts

  • Chafiq Benhida
  • Raúl E. CurtoEmail author
  • George R. Exner
Article

Abstract

We say that a weighted shift \(W_\alpha \) with (positive) weight sequence \(\alpha : \alpha _0, \alpha _1, \ldots \) is moment infinitely divisible (MID) if, for every \(t > 0\), the shift with weight sequence \(\alpha ^t: \alpha _0^t, \alpha _1^t, \ldots \) is subnormal. Assume that \(W_{\alpha }\) is a contraction, i.e., \(0 < \alpha _i \le 1\) for all \(i \ge 0\). We show that such a shift \(W_\alpha \) is MID if and only if the sequence \(\alpha \) is log completely alternating. This enables the recapture or improvement of some previous results proved rather differently. We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.

Keywords

Weighted shift Subnormal Completely monotone sequence Completely alternating sequence Moment infinitely divisible 

Mathematics Subject Classification

Primary 47B20 47B37 

Notes

Acknowledgements

The authors wish to express their gratitude to an anonymous referee for detecting an omission in the original statement of Theorem 3.1. The authors are also indebted to another anonymous referee for a detailed reading of the paper, and for several helpful suggestions which led to improvements in the presentation. Some of the proofs in this paper were obtained using calculations with the software tool Mathematica [22].

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Chafiq Benhida
    • 1
  • Raúl E. Curto
    • 2
    Email author
  • George R. Exner
    • 3
  1. 1.UFR de MathématiquesUniversité des Sciences et Technologies de LilleVilleneuve d’Ascq CedexFrance
  2. 2.Department of MathematicsThe University of IowaIowa CityUSA
  3. 3.Department of MathematicsBucknell UniversityLewisburgUSA

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