All Solutions to an Operator Nevanlinna–Pick Interpolation Problem

Abstract

The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna– Pick interpolation problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub–optimal Nehari problem and variations on the Nevanlinna– Pick interpolation problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorization lemma and state space calculations. A new feature is that we do not assume an additional stability assumption on our data, which allows us to view the Leech problem and a large class of commutant lifting problems as special cases. Although the paper has partly the character of a survey article, all results are proved in detail and some background material has been added to make the paper accessible to a large audience including engineers.

This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 90670 and 93406).