The problem of completing partial matrix functions as a classical interpolation problem

Abstract

In this paper we consider the problem of completing a {open_quotes}partial{close_quotes}matrix function of the form that is holomorphic inside the unit disc to a {open_quotes}complete{close_quotes} matrix function that is holomorphic inside the unit disc and has a nonnegative real part in that region. This problem can be treated as a generalization of the problem of weighted approximation in a uniform metric of an antianalytic function by analytic functions. This is the so-called classical interpolation problem. The simplest and best known representatives of the class of classical interpolation problems are the Nevanlinna-Pik problem and the step and trigonometric moments problems. The classical interpolation problems are usually discussed within classes of contracting analytic functions, functions with positive real (:or imaginary) parts, or related classes. A distinguishing characteristic of these problems is the fact that a criterion for their solvability is positivity of some kernel (matrix), and the set of solutions is described in terms of a linear fractional transformation. The positivity of some kernel (matrix), and the set of solutions is described in terms of a linear fractional transformation. The term {open_quotes}classical interpolation problem{close_quotes} prompted Akhiezer to name a monograph {open_quotes}The Classical Moments Problem{open_quotes}; this book was introduced to the mathematicans ofmore » the Odessa school about fifteen years ago, and has become a standard text. One of the most effective means of investigating such problems is a method based on the theory of extensions of operators in Hilbert spaces. The goal of the present paper is to include the completion problem in the theory of classical interpolation problems, which we have already reduced to an abstract schema, and to use this schema to provide criteria for solvability of problems on completion and description of solution sets. Of necessity, our presentation does not stand alone.« less