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](https://en.wikipedia.org/wiki/Mixing_(mathematics)) enough for this? Are such objects studied in the literature?



Operators "building" linear independant sets
Possibly I misunderstood your question, but it seems to me that an operator satisfying the condition should have $\{x, Tx\}$ linearly independent for nonzero $x$. Then the condition fails to be satisfied for $\{x\_i\}\_{i=1}^N=\{x,Tx\}$, because its image repeats a vector, so there are no operators $T$ satisfying the condition.


If you do not like contradiction by repetitions, you can pick $x\_1=x$, $x\_2=x+Tx$.


