Let $V$ be a vector space over a field $k$ (not necessarily finite-dimensional) and $T : V \to V$ a linear operator. Is there an accepted term for the following condition on $T$?
For any $v \in V$ the subspace $\text{span}(v, Tv, T^2 v, ...)$ is finite-dimensional, and $T$ is nilpotent on any such subspace.
For example, the differential operator $\frac{d}{dx}$ acting on $k[x]$ satisfies this condition but is not nilpotent.
Motivation: When $\text{char}(k) = 0$, this condition ensures that the exponential $e^T : V \to V$ is well-defined without giving $V$ any additional structure, since $e^T v$ is a finite sum for any particular $v$.