Two remarks on the set of recurrent vectors

Abstract

[EN] We solve in the negative two open problems, related to the linear and topological structure of the set of recurrent vectors, asked by Sophie Grivaux, Alfred Peris and the first author of this paper. Firstly, we show that there exist recurrent operators whose set of recurrent vectors is not dense lineable; and secondly, we construct operators which are reiteratively recurrent and cyclic, but whose set of reiteratively recurrent vectors is meager. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.The first author was supported by the Spanish Ministerio de Ciencia, Innovacion y Universidades, grant FPU2019/04094; by MCIN/AEI/10.13039/501100011033, Projects PID2019-105011GB-I00 and PID2022-139449NB-I00; and by the "Fundacio Ferran Sunyer i Balaguer". The second author is a Research Associate of the Fonds de la Recherche Scientifique - FNRS.López-Martínez, A.; Menet, Q. (2025). Two remarks on the set of recurrent vectors. Journal of Mathematical Analysis and Applications. 541(1). https://doi.org/10.1016/j.jmaa.2024.128686541