Moduli of Hankel operators and a problem of V. V. Peller and S. V. Khrushchev

Theorem. {\it Let $A$ be a bounded nonnegativ, selfadjoint operator such that $0\in\sigma(A)$, $\dim\operatorname{Ker}A=0$ or $\infty$, the operator $A|(\operatorname{Ker}A)^\bot$ is unitary equivalent to the operator of multiplication by $x$ in the space $L^2(\mu)$, where $\mu$ is the discrete measure. Then there exists a Hankel operator $H_\varphi$ such that the operator $A$ is unitarily equivalent to the operator $(H_\varphi^*H_\varphi)^{1/2}$.}