Suppose $T$ is a linear operator on some vector space $V$, and suppose $U$ is a $T$-invariant subspace of $V$. Does there necessarily exist a complement (a subspace $U^c$ such that $V=U\oplus U^c$) in $V$ which is also $T$-invariant?
I'm curious because I'm wondering if, given such $U$, it is always possible to decompose the linear operator $T$ into the sum of its restrictions onto $U$ and $U^c$, but I don't know if such a $T$-invariant $U^c$ exists.