Under invariant space

Question

Let $T:\mathbb C^n\rightarrow \mathbb C^n$ be un linear operator.

We know that if $W\in \mathbb{C}^{n}$ is $T$ invariant (i.e. $T(W)\subset W$), then $W$ is $p(T)$ for any polynomial $p\in \mathbb C$.

But is the converse true?

Answer 1

Yes,if $W$ is invariant by $P(T)$ where $P$ is any polynomiale, take $P(X)=X$, $P(T)(W)=T(W)\subset W$ since $X(T)=T$.

Answer 2

If the question is if $U$ being invariant under $p(T)$ for every polynomial $p$ implies that $U$ is invariant under $T$ then the answer is yes because $T$ itself has the form $p(T)$ where $p(x)=x$.

If the question is if $U$ being invariant under $p(T)$ for some polynomial $p\in\mathbb{C}[x]$ implies that $U$ is invariant under $T$ then it is false. For example, take $T:\mathbb{C^2}\to\mathbb{C^2}$ by $T(x,y)=(0,x)$. Then the subspace $U=span\{(1,0)\}$ is invariant under $T^2$ (because $T^2$ is just the zero operator) but not under $T$.