Exercise 11, § 5

Let $E =\ell^2(\mathbf{N})$ and let $(e_n)$ be the canonical basis of E. We define $u\in \mathscr{L}(E)$ by $u(e_i) =e_{i+1}$ for $i\in \mathbf{N}$. Let $F = E\oplus E$ and $v=u\oplus (u^*+ 2\cdot 1_E)\in \mathscr{L}(F)$. Let $\overline{v}=\pi (v)$ be the image of $v$ in the Calkin algebra $\mathscr{C}$alk(F).

(a) The endomorphism $v$ is a Fredholm endomorphism of index $-1$, and $v-2\cdot 1_E$ is a Fredholm endomorphism of index 1.

(b) Let $p= X(X-2)\in \mathbf{C}[X]$ and let S be the exponential spectrum of $\overline{v}$ (Exercise 5 of I, p. 173). We have $0\in p(S)$ but 0 does not belong to the exponential spectrum of $p(\overline{v})$. (Use Exercise 3.)