Resolvent Mapping is Analytic/Bounded Linear Operator

Theorem
Let $B$ be a Banach space.

Let $\map \LL {B, B}$ be the set of bounded linear operators from $B$ to itself.

Let $T \in \map \LL {B, B}$.

Let $\map \rho T$ be the resolvent set of $T$ in the complex plane.

Then the resolvent mapping $f : \map \rho T \to \map \LL {B, B}$ given by $\map f z = \paren {T - z I}^{-1}$ is analytic, and:


 * $\map {f'} z = \paren {T - z I}^{-2}$

where $f'$ denotes the derivative of $f$ $z$.