Riesz's theorem for orthogonal matrix polynomials
This paper extends to the matrix setting the classical results of Riesz and Nevanlinna connecting orthogonal polynomials, extremal solutions of a moment problem and the denseness of the set of polynomials in an \(L^2\) space. Namely, given a positive definite \(N \times N\) matrix of measures \(\nu\) and the corresponding sequence \(\{P_n\}\) of matrix polynomials, orthonormal with respect to \(\nu\), an infinite block Jacobi matrix \(J\) is constructed, whose entries are the coefficients of the three term recurrence relation satisfied by \(\{P_n\}\). The author considers only the completely indeterminate case, that is when both deficiency indices of \(J\) are \(N\). In this setting he defines an analogue of the Stieltjes (or Cauchy) transform of a matrix of measures and describes the image by this transform of all the solutions of the moment problem corresponding to \(\nu\). In doing so, a well-known theorem of Nevanlinna is extended. Finally, Riesz's theorem relating extremality of a measure \(\mu\) solving a moment problem with the denseness of the sequence of orthonormal polynomials in \(L^2(\mu)\) is generalized to the matrix framework, again under the hypothesis that the matrix moment problem is completely indeterminate.
scientific article; zbMATH DE number 1271887