On the Multiplicity of Singular Values of Hankel Operators Whose Symbol Is a Cauchy Transform on a Segment

We derive a result on the boundedness of the multiplicity of the singular values for Hankel operator, whose symbol is of the form F (z) := ∫ dλ(t ) z − t +R(z), where λ 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 ∫ t j qn(t )Q(t ) wn(t ) e q2 n(t ) dλ(t ) = 0, j = 0, . . . , n− s − 1, where Q is the denominator of R, s = deg(Q), e qn(z) = z n qn(1/z̄) is the reciprocal polynomial of q , and {wn} is the outer factor of an n-th singular vector ofHF .