Exercise 2, § 3

Let $p$ be a real number such that $1\leqslant p <+\infty$. Let $u$ be a diagonal endomorphism of $\ell^p_K(\mathbf{N})$. Now prove that $u$ is a Fredholm endomorphism if and only if it is a Riesz endomorphism. In particular, every diagonal Fredholm endomorphism of $\ell^p_K(\mathbf{N})$ has index zero.