I solved this exercise, that is the exercise 5.B.11 of Linear algebra done right, third edition of Axler
Let $V$ a complex vector space and $T\in\mathcal L(V)$, $p\in\mathcal P(\Bbb C)$ a polynomial and $\alpha\in\Bbb C$. Prove that $\alpha$ is an eigenvalue of $p(T)$ if and only if $\alpha=p(\lambda)$ for some eigenvalue $\lambda$ of $T$.
for any polynomial of degree at least one. My problem is that I cannot see if the result holds (vacuously or not) when the polynomial is constant or zero.
If $V$ would be finite-dimensional I know that $T$ have at least one eigenvalue. My problem is that if $V$ is infinite-dimensional and $T$ injective then I dont see clearly if $T$ must have necessarily at least an eigenvalue.
By the other side it doesnt seems correct to assert that $\alpha=p(\lambda)$ when such $\lambda$ doesnt exists, that is, the statement dont seems to hold vacuously.
Can someone help me to clarify this question?
P.S.: I dont know exactly what tags I must use for this question.