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A Subnormal Completion Problem for Weighted Shifts on Directed Trees

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Abstract

Given a directed tree and a collection of weights on a subtree, the subnormal completion problem is to determine whether the weights may be completed to the weights of an injective, bounded, subnormal weighted shift on the Hilbert space arising from the full tree. We study this problem (which generalizes significantly the classical subnormal completion problem for weighted shifts) both from a measure-theoretic point of view and in terms of initial data, for various classes of trees with a single branching point. We give several characterizations of when such a completion is possible. Considered also are connections with Stieltjes moment sequences, flatness of a completion, completions in which the resulting measures may be taken to be finitely atomic, and we provide a result showing that in certain circumstances the present completion problem is equivalent to a related classical completion problem.

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Acknowledgments

The authors take this opportunity to express their appreciation both for the support of their universities Bucknell University, Jagiellonian University and Kyungpook National University for visits materially aiding this collaboration, and to the Departments of Mathematics at which they have been guests for warm hospitality.

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Authors and Affiliations

  1. Department of Mathematics, Bucknell University, Lewisburg, PA, 17837, USA

    George R. Exner

  2. Department of Mathematics, Kyungpook National University, Daegu, 41566, Korea

    Il Bong Jung & Hye Yeong Yun

  3. Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, 30348, Kraków, Poland

    Jan Stochel

Authors
  1. George R. Exner
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  2. Il Bong Jung
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  3. Jan Stochel
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  4. Hye Yeong Yun
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Corresponding author

Correspondence to Jan Stochel.

Additional information

The research of the second author was supported by the Kyungpook National University Bokhyeon Research Fund, 2017. The research of the third author was supported by the NCN (National Science Center), decision No. DEC-2013/11/B/ST1/03613.

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Exner, G.R., Jung, I.B., Stochel, J. et al. A Subnormal Completion Problem for Weighted Shifts on Directed Trees. Integr. Equ. Oper. Theory 90, 72 (2018). https://doi.org/10.1007/s00020-018-2496-9

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  • DOI: https://doi.org/10.1007/s00020-018-2496-9

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