Exercise 6, § 4

Let $E =\ell^2(\mathbf{N})$ and $(e_n)_{n\in\mathbf{N}}$ be the canonical orthonormal basis of E. Let $u$ be the diagonal endomorphism of E such that $u(e_0) =e_0$ and $u(e_n) = 2e_n$ if $n\geqslant 1$. Now prove that $((u)) = 1$ and that $u+h$ is a Fredholm endomorphism for every $h$ such that $\|h\|<2$.