A Subnormal Completion Problem for Weighted Shifts on Directed Trees
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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.
KeywordsSubnormal operator Weighted shift on a directed tree Subnormal completion problem Flatness Stieltjes moment sequence
Mathematics Subject ClassificationPrimary 47B20 47B37 Secondary 05C20 44A60
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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