On Spaceability within Linear Dynamics

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 $T$.