On Spaceability within Linear Dynamics

We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \ (T: X X\), we introduce the set \ (Ω (T) \), consisting of all continuous linear operators \ (h: X X\) for which there exists a strictly increasing sequence \ ( (θₙ) ₙ\) of positive integers such that the set \ (\x X: ₍ ₓ^{⏗䂸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 \ (Ω (T) \). To analyze \ (Ω (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 \ (Ω (T) \) through the left-multiplication operator \ (LT\) acting on the algebra of bounded operators. Our approach combines two key ingredients: a refinement of A. López'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 \ (Ω (T) \), \ (APΩ (T) \), and for any countable c. s. a. by \ (T\).