Independence, infinite dimension, and operators

An interesting operatorial condition ensuring the equivalence between infinite dimension and linear independence of a sequence $(e_n)_{n \in \mathbb{N}}$ of vectors recently appeared in a paper made by O. Christensen and M. Hasannasab (proposition 2.3 of the paper "Frame properties of systems arising via iterative actions of operators"). It relies on the existence of an operator $T$ sending $e_n$ to $e_{n+1}$ for all $n \in \mathbb{N}$. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then show that, in the restricted context of vector spaces, the result can at most be generalized to families $(e_i)_{i \in I}$ indexed by a countable set $I$ and maps $\phi : I \to I$ that are conjugate to the successor function $s : n \mapsto n+1$ defined on $\mathbb{N}$, at least if we want to preserve a condition like $T(e_i)=e_{\phi(i)}$ for all $i \in I$. We finally prove a tentative generalization of the result, where we replace the condition $T(e_i)=e_{\phi(i)}$ for all $i \in I$ with a more sophisticated one, and to which we have not managed to find a new application yet.