Some remarks on the operator of Foias and Williams

Abstract:

In this paper we study the Foias-Williams operator \begin{equation*}T(H_{g})=\begin {pmatrix}S^{*} & H_{g} \ 0 & S \end{pmatrix} \end{equation*} where $g\in L^{\infty }$, and $H_{g}$ is a Hankel operator with symbol $g$. We exhibit a relationship between the similarity of $T(H_{g})$ to a contraction and the rate of decay of $\{|g_{n}|\}_{n=0}^{\infty }$, the absolute values of the Fourier coefficients of the symbol $g$.