Let $E$ be a separable Banach space and let $T\in L(E,E)$. 

Is there a condition on $T$ ensuring that:
$$
\mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow
\{T(x_n)\}_{n=1}^N\cup \{x_n\}_{n=1}^N \mbox{ is a independent in $E$}?
$$

Is $T$ [mixing][1] enough for this?  Are such objects studied in the literature?


  [1]: https://en.wikipedia.org/wiki/Mixing_(mathematics)