Integral Equations and Operator Theory

, Volume 9, Issue 2, pp 155-203

First online:

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: Parametrization of the set of all solutions

  • Joseph A. BallAffiliated withDepartment of Mathematics, Virginia Polytechnic Institute
  • , J. William HeltonAffiliated withDepartment of Mathematics, University of California

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We consider a general matrix version of a Pick-Loewner interpolation problem on the closed unit disk. Solutions are allowed to have a finite numberl of free poles in the open disk. We show that the smallestl for which a solution to the problem exists is the number of negative eigenvalues of an appropriately defined “Pick matrix,” and for this value ofl we obtain a linear fractional map parametrization of the class of all solutions. The idea is to adapt the Grassmannian approach involving Krein space geometry and invariant subspace representations of the authors; this was successful previously for the case where all interpolating points are inside the disk. Also an appendix includes an errata to earlier work together with simplified proofs.