ACTA issues

Weak orbit-transitivity on Hilbert space

Ciprian Foias, Carl Pearcy, Lidia Smith

Acta Sci. Math. (Szeged) 76:1-2(2010), 155-164
61/2008

 Abstract. This note is concerned with weakly hypercyclic vectors and operators and weakly orbit-transitive operators (definitions below). We show that, given a sequence $\{x_{n}\}$ of vectors with $\|x_{n}\|\to\infty$ and $0\in\{x_{n}\} ^{^{\underline{w}}}$, there exists another sequence $\{w_{n}\}$ with approximately equal growth rate that is weakly dense. This complements a result of V. Kadets [11]. Then we apply Kadets' theorem, together with others used previously [10], to show that certain classes of operators consist entirely of non-weakly-orbit-transitive operators, thereby generalizing the results of [10]. Along the way we show that K. Ball's complex-plank theorem [2] is equivalent to a (slightly stronger) version of an old theorem of Beauzamy [3]. AMS Subject Classification (1991): 47A15, 47A16, 47B37 Keyword(s): orbit-transitive, hypertransitive Received November 24, 2008, and in revised form March 11, 2009. (Registered under 61/2008.)