Recurrent subspaces in Banach spaces

TL;DR

This paper investigates the conditions under which operators on Banach spaces have recurrent subspaces, revealing a deep connection between recurrence, hypercyclicity, and spectral properties.

Contribution

It establishes new criteria linking recurrent subspaces to spectral conditions and extends known results to real Banach spaces, unifying hypercyclicity and recurrence theories.

Findings

Recurrent subspaces exist under certain spectral conditions.

Having a recurrent subspace is equivalent to the essential spectrum intersecting the unit disc.

Weakly-mixing operators have hypercyclic subspaces if and only if they have recurrent subspaces.

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 disc; 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 recurrence spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals.