Exercise 23, § 2

Let $u$ be an endomorphism of E and let A be the closed unital subalgebra of $\mathscr{L}(E)$ generated by $u$.

If $v$ is an endomorphism of E such that $u\circ v-v\circ u$ belong to A, then $u\circ v-v\circ u$ is quasinilpotent. In particular, one has $u\circ v-v\circ u\not = 1_E$ if E is nonzero. (Use Exercise 10 of I, p. 167; note that the result is valid for every Banach space E.)