Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

By George Costakis, Ioannis Parissis. 2010

59 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/97e31f49626db771dbcc33660632c70506ecddd8

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In Algebra and Number Theory

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Frequently recurrent operators (2020, background): “As pointed out by the referee, there is an important notion of (topological) multiple recurrence studied for linear operators in [20].”
Recurrent Linear Operators (2013, background): “The recurrent properties of multiplication operators on Hilbert spaces of analytic functions as above are contained in the following theorem, proved in [15]:”