On Spaceability within Linear Dynamics (2509.06156v1)

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).