Here is one of the practice questions that I have been trying from my C*-algebras exercises.
Suppose that $T\in L(X)$ and $f\in H(T)$. Prove that $f(T^*) = f(T)^*$
In this situation $H(T)$ represents all functions that are analytic around a neighborhood of $\sigma(T)$. I tried to use continuous functional calculus here, but that is only applicable for normal elements. So, I feel like I should be using the analytic functional calculus. But I can't make this work since $\sigma(T)$ need not be equal to $\sigma(T^*)$.
Any hints on how to proceed will be greatly appreciated :)