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 $\lim_{z\to\infty} \|f(z)\|_* = 0$.

Proof
Pick $z \in \Bbb C$ with $|z| > 2\|T\|_*$.

Then $\|T/z\|_* = \|T\|_*/|z| < 1/2$ by Operator Norm is Norm. So we have:

Taking limits of both sides as $|z| \to \infty$, we get $\|f(z)\|_* \to 0$.