I need help from someone to solve this problem.
Given a bounded sequence $(\lambda_n)$ in $\mathbb С$ define an operator $S$ in $B(\ell_2)$ by $S(x_1) = 0$ and $S(x_n) = \lambda_n x_{n-1}$ , $n > 1$, for $x = (x_n) \in \ell_2$. Find the polar decomposition of $S$, and characterize those sequences $\{\lambda_n\}$ in $\ell_\infty$ for which $S$ is compact.
Write $\lambda_n=|\lambda_n|e^{i\theta_n}$ and $S=UP$, where $(Px)(n)=|\lambda_n| x(n)$, and $U(x)(n)=e^{i\theta_n}x(n-1)$ if $n\geqslant 1$ and $0$ if $n=0$. Then $P$ is non-negative definite, $U$ is unitary.
Show that $S$ is compact if and only if $\lim_{n\to +\infty}\lambda_n=0$. It's done here.