Invariant subspace of two operators T and U when $TU=UT$

Question

$V$ is a finite dimensional vector space. Suppose I have tow linear operators $T,U$ on $V$ such that $TU=UT$.

We know that the range of $T$ and $\ker T$ is a invariant subspace of $U$. This motivates me to investigate whether all the invariant subspace of $T$ are invariant subspace of $U$? If not provide a counterexample.

I searched for examples with the known operators ( like projection along $x$ axis and projection along $y$ axis). But I didn't find a example in which the above statement is false. So please give me some hint to solve this problem.

Thanks in advance.

Answer 1

Take $T$ to be the identity. Then every subspace is $T$-invariant. But in general, $IU=UI$ and not every subspace is $U$-invariant.