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Journal of Operator Theory

Volume 61, Issue 2, Spring 2009  pp. 239-251.

On the multiplicity of singular values of Hankel operators whose symbol is a Cauchy transform on a segment

Authors M. Yattselev
Author institution: INRIA, Projet APICS, 2004 route des Lucioles - BP 93, Sophia-Antipolis, 06902, France

Summary:  We derive a result on the boundedness of the multiplicity of the singular values for Hankel operator, whose symbol is of the form \begin{equation*} F(z):=\int\frac{\mathrm{d}\mes(t)}{z-t}+R(z), \end{equation*} where $\mes$ is a complex measure with infinitely many points in its support which is contained in the interval $(-1,1)$, and whose argument has bounded variation there, while $R$ is a rational function with all its poles inside of the unit disk. For that we use results on the zero distribution of polynomials satisfying the orthogonality relations of the form \begin{equation*} \int t^jq_n(t)Q(t)\frac{w_n(t)}{\widetilde q_n^2(t)}\mathrm{d}\mes(t)=0, \quad j=0,\ldots,n-s-1, \end{equation*} where $Q$ is the denominator of $R$, $s\!=\!\deg(\!Q\!)$, $\widetilde q_n(\!z\!)\!\!=\!\!z^n\overline{q_n\!(1\!/\overline z)}$ is the reciprocal polynomial of $q$, and $\{\!w_n\!\}$ is the outer factor of an $n$-th singular vector of $\ho_F$.

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