Exercise 11, § 2

Let H be a Hilbert space of countable type and $(f_n)_{n\geqslant 1}$ an orthonormal basis of H. Let A be the normed algebra $\mathscr{L}(H)$. Let $x\in A$ be the element defined by $x(f_i) = 2^{-i}f_{i+1}$. Prove that $x$ is quasi–nilpotent, but not nilpotent.