Why is the spectrum of a shift operator the closed unit disk?

Consider the following text from Murphy’s: "$C^*$-algebras and operator theory":

In example 2.3.2, why is $sigma(u) = Bbb{D}$ (= the closed unit disk)?

I can see that $sigma(u) subseteq Bbb{D}$ and $sigma(u^*) = Bbb{D}.$

Thanks in advance!