Chaotic behavior of the Riesz transforms for Hermite expansions

We prove a characterization for hypercyclic and chaotic unbounded unilateral weighted shifts of order p. As applications we obtain that the natural derivatives associated to Hermite expansions are chaotic. On the other hand, the corresponding Riesz transforms are not hypercyclic and even more they are a kind of border line operator which separates the chaotic behavior. © 2007 Elsevier Inc. All rights reserved