TITLE: Polynomial and spectrum of a Operator .
QUESTION [8 upvotes]: I need help with the following question : 
Let us consider $X$ to be a $\mathbb C  $ Banach space . Let $T \in B(X)$ ie. $T$ is a continuous linear map from $X$ to $X$ . define $$P(T)=\sum_{k=o}^{n}a_kT^k$$ with $T^0=id_X$ (identity) . 
The claim is that the spectrum of $T : \sigma(T) \subset\{\lambda \in \mathbb C : P(\lambda)= 0\}$ 
Here $P$ is a polynomial such that $P(T) =0$  . 
Thanks.

REPLY [8 votes]: Let me expand on the setting. Let $\text{Hol}(T)$ be the set of all the functions holomorphic in a neighborhood of $\sigma(T)$. The Riesz functional calculus now says that indeed if I have the polynomials $p$, that $p(T)$ is as expected - of course, for polynomials this is trivial.
Additionally, we have the Banach algebra of bounded operators on $X$, and:
Spectral mapping theorem: If $T$ is an element of a Banach algebra $\mathcal A$ and $f$ is holomorphic in a neighborhood of $\sigma(T)$, then
$$\sigma(f(T)) = f(\sigma(T)).$$
If we thus select such a polynomial $p$ such that $p(T) = 0$, we get $p(\sigma(T)) = 0$.