Exercise 23, § 4

Tag 0357, permanent. Cite it as [Bourbaki, Tag 0357].

Spectral Theories, IV, p. 352

Let $E =\ell^2_{\mathbf{C}}(\mathbf{N})$ and let $u$ be the endomorphism of E defined by

u((x_i)_{i\in\mathbf{N}}) = (0, x_1, . . . , x_n, . . .)

for all $(x_i)\in E$ .

a) The spectrum of $u$ is the closed unit disk in $\mathbf{C}$ .

b) The residual spectrum of $u$ is the open unit disk in $\mathbf{C}$ .

c) The essential spectrum of $u$ is the unit circle in $\mathbf{C}$ .

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