the Hilbert algebra and spectrum

Question

Let‎ ‎$ H = \ell‎^{2}(\mathbb n) $, and let $‎S : H ‎‎\longrightarrow‎‎ H$‎‎ be the unilateral shift $$ S(x‎_{‎1‎}‎, x‎_{‎2‎}‎, \ldots) = (0, x‎_{‎1‎}‎ , x‎_{‎2‎}‎,\ldots) . $$‎‎

Show ‎that:

‎>$‎S ‎\in ‎B(H)$,

$ ‎‎\| S‎\|= 1 $,

$‎\sigma(S) = \{ ‎\lambda \in ‎\mathbb{C} : ‎\mid\lambda\mid\leqslant 1\}‎‎$‎‎

Answer 1

Show that $S^*S=I $. For the spectrum, show that any $\lambda $ with $|\lambda|<1$ is an eigenvalue of $S^*$.