On Spaceability within Linear Dynamics

Abstract

We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \(T:X \to X\), we introduce the set \(\Omega(T)\), consisting of all continuous linear operators \(h:X \to X\) for which there exists a strictly increasing sequence \((\theta_n)_n\) of positive integers such that the set \(\{x \in X : \displaystyle{\lim_{n \rightarrow \infty} T^{\theta_n}x = h(x)}\}\) is dense in \(X\). Within this framework, two classical phenomena--the existence of hypercyclic and recurrent subspaces in separable infinite-dimensional complex Banach spaces--emerge as instances of a common underlying structure described by \(\Omega(T)\). To analyze \(\Omega(T)\), we introduce the notion of collections simultaneously approximated (c.s.a.) by \(T\), and show that every maximal c.s.a. is an SOT-closed affine manifold. For quasi-rigid operators on separable Banach spaces, we establish the existence of a unique maximal c.s.a. containing the identity operator. Furthermore, we examine \(\Omega(T)\) through the left-multiplication operator \(L_T\) acting on the algebra of bounded operators. Our approach combines two key ingredients: a refinement of A. L\'opez's technique on recurrent subspaces for quasi-rigid operators, and a common dense-lineability result obtained by the first author and A. Arbieto. These tools yield new spaceability results for the sets \(\Omega(T)\), \(\mathcal{AP}\Omega(T)\), and for any countable c.s.a. by \(T\).