Expressing transformations as polynomials of $T$ over $T$-cyclic space

Question

I'm reading a book about linear algebra (Lineare Algebra, Bröcker) and it contains the following exercise:

Let $T \in End_{K}(V)$ be nilpotent and V be a $T$-cyclic vector space ($V = \langle x,Tx,T^2x,...\rangle$ for some $x\in V$).
Proof that every $G \in End_{K}(V)$ with $GT = TG$ can be written as a polynomial of $T$.

I can't figure out how to use the fact that $TG = GT$. I'm grateful for any hints.
($End_K$ are the Endomorphisms over the field $K = \mathbb{C},\mathbb{R}$)

Here's a hint: you might find it easier to prove using matrices. It is equivalent to show that if $$ N = \pmatrix{0\1&0\&\ddots&\ddots\&&1&0} $$ and $P$ is a matrix with $PN = NP$, then $P$ must be lower-triangular and Toeplitz. — Ben Grossmann, Jul 21 '21 at 18:19
@plshelp In fact your question is easier than the one in the link (which proves the opposite direction). — Zhanxiong, Jul 21 '21 at 18:41

score 2 · Accepted Answer · answered Jul 21 '21 at 19:00

The proof of this is easy, and doesn't require $T$ to be nilpotent.

Take $G \in \operatorname{End}_K(V)$ with $GT=TG$. Then, as every element of $V$ is a linear combination of $v,Tv,T^2v,\dots$ we have that $Gv = a_0v+a_1Tv+\cdots+a_nT^nv$ for some $n \in \mathbb Z^+$ and $a_0,\dots,a_n \in K$. Thus, for $p = a_0+a_1x+\cdots+a_nx^n \in K[x]$ we then have $Gv = p(T)v$, and by induction $GT^kv = p(T)T^kv$ for every $k \in \mathbb Z^+$ (for example, $$GTv = TGv = Tp(T)v = p(T)Tv$$ and $GT^2v = TGTv = Tp(T)Tv = p(T)T^2v$). Thus $G$ and $p(T)$ agree on a generating set for $V$, so $G = p(T)$.

Expressing transformations as polynomials of $T$ over $T$-cyclic space

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