TITLE: the Hilbert algebra and spectrum
QUESTION [0 upvotes]: 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 $,
$‎\sigma(S) = \{ ‎\lambda \in ‎\mathbb{C} : ‎\mid\lambda\mid\leqslant 1\}‎‎$‎‎

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