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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 enough for this? Are such objects studied in the literature?

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  • $\begingroup$ I don't think that chaotic is sufficient. There are chaotic weighted shift operators on $\ell^2$. For the standard unit vectors (sequences) $e_n$ the image $T(e_2)$ is a multiple of $e_1$. $\endgroup$
    – Jochen Wengenroth
    Commented Jun 16, 2021 at 8:00

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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 $x_1=x$, $x_2=x+Tx$.

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