Resolvent Mapping Converges to 0 at Infinity

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 such that $\ds \lim_{z \mathop \to \infty} \norm {\map f z}_* = 0$.

Proof
Pick $z \in \Bbb C$ such that $\size z > 2 \norm T_*$.

By Operator Norm is Norm:
 * $\norm {\dfrac T z}_* = \dfrac {\norm T_*} {\size z} < \dfrac 1 2$

Hence:

Taking limits of both sides as $\size z \to \infty$:
 * $\norm {\map f z}_* \to 0$