Abstract:  

Motivated by questions concerning ``better than subnormal'' weighted shifts on Hilbert space, we define a class of weighted shifts whose weight sequences are geometric in a certain sense.  These shifts [respectively, their moment sequences] exhibit a variety of properties related to subnormality, infinite divisibility, and complete hyperexpansitivity of operators [respectively, completely monotone, log complete monotone, and completely alternating sequences];  the moment sequences possess as well a useful and important signed representing measure. We use these measures, as well as interpolating functions of familiar types (Bernstein and completely monotone) as tools in the study, and obtain some function theory results.  Other general results (and questions) follow, including a broadening of some classical questions in moment theory.  [Joint work with Chafiq Benhida and Ra\'{u}l Curto]

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