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

$$V$$ 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.

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