Say we're given a diagonalizable endomorphism $T: V \to V$, $V$ an $n$-dimensional Vectorspace over some Field $F$. Let $U \subsetneq V$ be T-cyclic with generator $u \in U$. I want to prove that $U$ is an eigenspace. (I suspect it is a 1-dimensional Eigenspace).I'm not sure how to proceed. It is easy to show the other direction (any (must it be 1-dimensional? I suspect yes, but not sure) eigenspace is $T$-cyclic), but this direction is harder.
edit: what if I strengthen the hypothesis and say that while $U$ itself is $T$-cyclic, it is not decomposable into proper $T$-cyclic subspaces. Must $U$ then necessarily be an eigenspace (of dimension 1)?