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Moment infinitely divisible weighted shifts. (English) Zbl 07032878
Summary: We say that a weighted shift \(W_\alpha \) with (positive) weight sequence \(\alpha : \alpha_0, \alpha_1, \dots \) is moment infinitely divisible (MID) if, for every \(t > 0\), the shift with weight sequence \(\alpha ^t: \alpha_0^t, \alpha_1^t, \dots \) is subnormal. Assume that \(W_{\alpha}\) is a contraction, i.e., \(0 < \alpha_i \leq 1\) for all \(i \geq 0\). 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.

MSC:
47B20 Subnormal operators, hyponormal operators, etc.
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Software:
Mathematica
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