Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

Abstract

Let T be a bounded linear operator acting on a complex Banach space X and (λ_n) a sequence of complex numbers. Our main result is that if |λ_n|/|λ_{n+1}| \to 1 and the sequence (λ_n T^n) is frequently universal then T is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemerédi's theorem in arithmetic progressions. We show that the previous assumption on the sequence (λ_n) is optimal among sequences such that |λ_n|/|λ_{n+1}| converges in [0,+\infty]. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.

18 pages; to appear in Math. Scand., this second version of the paper is significantly revised to deal with the more general case of a sequence of operators (λ_n T^n). The hypothesis of the theorem has been weakened. The numbering has changed, the main theorem now being Th. 3.8 (in place of Proposition 3.3). The changes incorporate the suggestions and corrections of the anonymous referee