Can I prove the spectral mapping theorem fore non-zero constant polynomials?
If $X$ is a complex Banach space and $p$ is any polynomial with complex coefficients and $A$ then how can I show that $\sigma(p(A)) = p(\sigma(A))$?
Basically, I want to show that $\sigma(\alpha I) = \{\alpha\}$ for any complex number $\alpha$?
I have shown that $\{\alpha\}$ $\subset$ $\sigma(\alpha I)$.
How to show the other way round?