So here is the problem I'm working on with.
''Let $V$ be a finite dimensional vector space over $\mathbb{C}$ and let $T: V \rightarrow V$ be a linear operator. Suppose that for every subspace $U$ of $V$, if $U$ satisfies $T(U) \subset U$, then there exist a subspace $W$ of $V$ such that $T(W) \subset W$ and $V=U \oplus W$. Prove that $T$ is diagonalizable.''
I think using mathematical induction on the dimension of $V$ might solve the problem but I can't come up with any significant ideas to solve this. Maybe spectrum theorem or Shur's theorem might help but I just can't figure it out.