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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 $v$.

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Is this somehow different from just saying that for any $v \in V$, there exists $n \in \mathbb{N}$ such that $T^nv = 0$? – goblin Jun 15 at 20:22
up vote 4 down vote accepted

The standard name is locally nilpotent. Thus one hears about locally nilpotent derivations, for example, like $\frac{\mathrm d}{\mathrm dt}$ in $k[t]$.

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Great! This does in fact seem to be established terminology; I googled "locally nilpotent" but didn't think to google "locally nilpotent derivation" (which is of course what I am actually interested in). – Qiaochu Yuan Feb 19 '11 at 17:03

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