Let X be a complex Banach space and $T \in L(X)$. Spectral mapping theorem states that for any complex polynomial p $$\sigma(p(T)) = p(\sigma(T)). $$ When I was analysing the proof of this theorem I managed to show that for continuous spectrum $\sigma_C$, residual spectrum $\sigma_R$ and approximate point spectrum $\sigma_{AP}$ we get one side inclusions of this theorem, namely $$ \sigma_i (p(T)) \subset p(\sigma_i(T)),$$ for $i \in \{AP,R,C\}$.
I managed to show that for $\sigma_C$ this inclusion is strict, by studying spectrum of a multiplication operator on $\ell^2$ (in my example $\lambda \in \sigma_C(T)$, but $\lambda^2 \in \sigma_P(T^2)$. I would like to find similar examples for residual spectrum, and perhaps for approximate point spectrum, but unfortunately here I'm stuck. I feel like this inclusion will be strict for residual spectrum, but I'm not that confident about approximate point spectrum (I didn't manage to prove the opposite inclusion for this part of spectrum thought).
I would appreciate any help regarding this!
Just to be clear, I consider the following definitions (from Kreyszig's book): $$\sigma_C := \{\lambda \in \mathbb{C}: T - \lambda \text{ is 1-1}, \operatorname{range}(T - \lambda) \text{ is dense}, (T - \lambda)^{-1} \text{ is unbounded} \},$$ $$ \sigma_R := \{\lambda \in \mathbb{C}: T - \lambda \text{ is 1-1}, \operatorname{range}(T - \lambda) \text{ is not dense}\},$$
$$\sigma_{AP} := \{\lambda \in \mathbb{C}: \lVert (T-\lambda)x_n\rVert \rightarrow 0 \text{ for some } (x_n) \subset X \text{ such that } \lVert x_n \rVert = 1, n \in \mathbb{N}\}.$$