homomorphism or not

Question

Let $T$ be a bounded operator on $H$ and fix a vector $x\in H$. Define $f$ on the space of polynomials in $T$ by $f(p(T))=p(x)$. Is $f$ a homomorphism? Initally I thought it obvious but the subtelty is that 'multiplication' for polynomials in $T$ is composition.

Answer 1

No. A homomorphism should map $0$ to $0.$ But $p(T)=0$ does not imply $p=0.$ Take e.g. $T=T^*$ with finite spectrum $\{\lambda_1,\dots,\lambda_n\}$ and $p(x)=(x-\lambda_1)\dots(x-\lambda_n).$

Note, that $p(x)\mapsto p(T)$ is a homomorphism $\mathbb C[x]\to B(H).$