Exercise 10, § 5

Let E be a separated locally convex space, $u$ and $h$ commuting elements of $\mathscr{L}(E)$. Suppose that there exists an integer $n$ such that $h^n$ is a compact endomorphism.

a) If $u$ is a Fredholm endomorphism, then $u-h$ is a Fredholm endomorphism.

b) If $u$ is a Riesz endomorphism, then $u-h$ is a Riesz endomorphism.