Moment Infinitely Divisible Weighted Shifts

We say that a weighted shift W α with (positive) weight sequence α : α 0 , α 1 , … is moment infinitely divisible (MID) if, for every t > 0 , the shift with weight sequence α t : α 0 t , α 1 t , … is subnormal. Assume that W α is a contraction, i.e., 0 < α i ≤ 1 for all i ≥ 0 . We show that such a shift W α is MID if and only if the sequence α 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.