Spectral theorem for compact operators on Banach space.

Question

Let $X$ be a Banach space. Let $A$ be a compact operator on $X$ and let's denote $\sigma(A)$ is spectrum of operator $A$. Let $f$ be holomorphic function in some neighbourhood of $\sigma(A)$

Out theacher sais that then there is exists a map $f \to f(A) \in B(X)$ (compact operators on $X$), such as :

1)$\alpha f_1(A) + \beta f_2(A) = (\alpha f_1 + \beta f_2)(A)$

2)$f_1 (f_2(A)) = f_1(A)\cdot f_2(A)$

3)$f(z) = \sum a_n z^n$ , then $f(A) = \sum a_n A^n$.

It looks good, but in Hilbert space, not in Banach. Is it true in general? Maybe I can read about it somewhere?

Answer 1

For introductory notes read Wikipedia:

Holomorphic Functional Calculus

For follow-up, I would suggest, Perturbation Theory, by Tosio Kato.