Exercise 12, § 2

Let H be a Hilbert space of countable type and $(f_n)_{n\geqslant 1}$ an orthonormal basis of H. The numbers $(\alpha_m)_{m\geqslant 1}$ are defined by $\alpha_m=e^{-k}$ if $m$ is the product of $2^k$ by an odd number. a) Let $x\in \mathscr{L}(H)$ be defined by $x(f_m) =\alpha_mf_{m+1}$ for $m\geqslant 1$. Prove that $x$ is not quasi-nilpotent. (Observe that

$\|x^n\|=$ sup$_m(\alpha_m\alpha_{m+1}. . . \alpha_{m+n-1})$,

and evaluate $\alpha_1\alpha_2. . . \alpha_{2^t-1}.)$

b) Let $x_k\in \mathscr{L}(H)$ be defined by $x_k(f_m) = 0$ if $m$ is the product of $2^k$ by an odd number, and $x_k(f_m) =\alpha_mf_{m+1}$ otherwise. Prove that $x_k$ is nilpotent and that $(x_k)$ converges to $x$ when $k$ tends to $+\infty$.