Spectral Mapping Theorem

Question

Spectral mapping theorem is as follows:
https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf

Is Spectral mapping theorem true for point spectrum ?

Answer 1

Your reference is talking about matrices. Which spectral mapping theorem are you interested in? For example, here's one version (see e.g. Rudin, "Functional Analysis", theorems 10.28 and 10.33):

Suppose $T$ is a bounded linear operator on the complex Banach space $X$, $\Omega \subseteq \mathbb C$ open with $\sigma(T) \subset \Omega$, and $f$ analytic in $\Omega$. Then the holomorphic functional calculus defines $f(T)$ such that, among other things (where $\sigma_p$ denotes point spectrum):

1. $f(\sigma(T)) = \sigma(f(T))$
2. $f(\sigma_p(T)) \subseteq \sigma_p(f(T))$
3. If there is no connected component of $\Omega$ on which $f$ is constant, then $f(\sigma_p(T)) = \sigma_p(f(T))$.

However, without that restriction the statement would be false. Consider e.g. an operator $T$ which has no point spectrum, and let $f$ be identically $0$. Then $\sigma_p(f(T)) = \sigma_p(0) = \{0\}$, but $f(\sigma_p(T)) = \emptyset$.