Let $\displaystyle \mathcal{A}$ be the space of all bounded analytic functions in the open unit disk in the complex plane.

My question is why is die spectrum the closure of the range of $\displaystyle f\in\mathcal{A}$ and not just the normal range?

$\displaystyle \sigma(f):=\{\lambda\in \mathbb{C}: (\lambda 1 -f) \notin Inv(\mathcal{A}) \}$

or should I look to the resolvant to find my answer?