Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

Details Szemeredi's theorem, frequent hypercyclicity and multiple recurrence Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

2010-08-24

arxiv.org/abs/1008.4017v2

Mathematics

Functional Analysis

18 pages; to appear in Math. Scand., this second version of the paper is significantly revised to deal with the more general c

Scientific paper

Let T be a bounded linear operator acting on a complex Banach space X and (\lambda_n) a sequence of complex numbers. Our main result is that if |\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is frequently universal then T is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemer\'edi's theorem in arithmetic progressions. We show that the previous assumption on the sequence (\lambda_n) is optimal among sequences such that |\lambda_n|/|\lambda_{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.

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