Questions in linear recurrence: From the $T\oplus T$-problem to lineability

Abstract

We study, for a continuous linear operator $T$ on an F-space $X$, when the direct sum operator $T\oplus T$ is recurrent on $X\oplus X$. In particular: we establish, for recurrence, the analogous notion to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each infinite-dimensional Banach space, solving the $T\oplus T$-recurrence problem in the negative way. The quasi-rigidity notion is closely related to the dense lineability of the set of recurrent vectors, and using similar conditions we study the lineability and dense lineability properties for the set of $\mathcal{F}$-recurrent vectors. This document has been split into two already published papers: Part I - Questions in linear recurrence I: The $T\oplus T$-recurrence problem. Analysis and Mathematical Physics, Volume 15, article number 1, (2025). https://doi.org/10.1007/s13324-024-00999-8 Part II - Questions in linear recurrence II: Lineability properties. Banach Journal of Mathematical Analysis, Volume 19, article number 61, (2025). https://doi.org/10.1007/s43037-025-00448-z