Recurrent Subspaces in Banach Spaces

Abstract

Abstract We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular, we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space, we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disk, and we extend the previous result to the real case. As a consequence, we obtain that a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrent spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals.