Szemeredi's theorem, frequent hypercyclicity and multiple recurrence
Abstract
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.
Cite
@article{arxiv.1008.4017,
title = {Szemeredi's theorem, frequent hypercyclicity and multiple recurrence},
author = {George Costakis and Ioannis Parissis},
journal= {arXiv preprint arXiv:1008.4017},
year = {2013}
}
Comments
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 (\lambda_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