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

Question

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!

Answer 1

$$ \lambda \in \sigma(u) \iff \overline{\lambda} \in \sigma( u^*).$$