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arXiv · 2006.11428

Frequently recurrent operators

Antonio Bonilla·Karl-G. Grosse-Erdmann·Antoni López-Martínez·Alfred Peris
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Abstract

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari- and L\'eon-M\"uller-type theorems for $\mathcal{F}$-recurrence under very weak assumptions on the Furstenberg family $\mathcal{F}$. This allows us, as a by-product, to deduce Ansari- and L\'eon-M\"uller-type theorems for $\mathcal{F}$-hypercyclicity.

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Antonio Bonilla, Karl-G. Grosse-Erdmann, Antoni López-Martínez, Alfred Peris. 2020-06-19. Frequently recurrent operators. https://doi.org/10.1016/j.jfa.2022.109713

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