$\DeclareMathOperator\C{C}\DeclareMathOperator\HC{HC}$Definitions:
Let $T:X\rightarrow X$ be a bonded linear operator on a separable (infinite-dimensional) Banach space and define the sets:
- $ \HC(T)\triangleq \left\{ x \in X:\, \overline{\{T^n(x)\}_{n \in \mathbb{N}}} = X \right\}, $
- $ \C(T)\triangleq \left\{ x \in X:\, \overline{\operatorname{span}\{T^n(x)\}_{n \in \mathbb{N}}} = X \right\}. $
*When $\HC(T)\neq \emptyset$, then $T$ is called hypercylic.*
Facts:
It is known (see Theorem 8.13 of Grosse-Erdmann and Manguillot - Linear chaos) that every separable Banach space admits a hypercyclic operator $T$. Moreover, by the Birkhoff's Transitivity Theorem, it is known that for such an operator $T$, the set $\HC(T)$ is a dense $G_{\delta}$ subset of $X$ not containing the origin.
Question:
Suppose that $T$ is hypercyclic. What do we known about $\C(T)$ other than the fact that $\HC(T)\subseteq \C(T)$ (and in particular $\C(T)\cup\{0\}$ contains a dense linear subspace)?
More specifically: How much "bigger" is $\C(T)$ than $\HC(T)$?
-I leave the interpretation of "bigger" up to the reader.
