Exercise 15, § 4

Tag 034Z, permanent. Cite it as [Bourbaki, Tag 034Z].

Spectral Theories, IV, p. 348

Let $E$ be the complex Hilbert space $\mathbf{C}^2$ and let $u\in \mathscr{L}(E)$ be the endomorphism such that $u(ae_1+be_2) =be_1$ .

a) Now prove that the spectrum of $u$ consists of 0 alone and that, for every $\lambda \in \mathbf{C}^*$ , one has the formula

\|R(u, \lambda )\|^2=\frac{2}{1 + 2|\lambda|^2 -\surd\overline{1 + 4|\lambda|^2}}

Deduce the form of the pseudospectrum PSp $_{\varepsilon}(u)$ for every $\varepsilon >0$ .

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