V as a direct sum of T-Invariant subspaces

Question

The primary decomposition theorem shows that there exist T-Invariant subspaces $U,W$ such that $V = U \oplus W$
But i am curious, if you are given any arbitary T-Invariant subspace of $V$, say $Y$, is the fact above enough to say that there exists some other T-Invariant subspace $Y'$ such that $V = Y \oplus Y'$

Answer 1

Hint Consider $V = \Bbb R^2$. Any proper subspace $0 \subsetneq Y \subsetneq V$ is $1$-dimensional, so if it is $T$-invariant for some $T$, it is a (subset of an) eigenspace of $T$.

Additional hint The linear transformation $T : (x, y) \mapsto (y, 0)$ $T$ only has one eigenspace, namely, $\Bbb R \times \{ 0 \}$.