All solutions to an operator Nevanlinna-Pick interpolation problem

TL;DR

This paper provides a complete explicit description of all solutions to the operator Nevanlinna-Pick interpolation problem with a positive Pick operator, using co-isometric realizations and without stability assumptions.

Contribution

It offers a new, detailed solution framework for the operator Nevanlinna-Pick problem that encompasses the Leech problem and commutant lifting problems without stability constraints.

Findings

Explicit formulas for all solutions under positive Pick operator

Solution techniques based on co-isometric realizations and Douglas factorization

Includes background and detailed proofs accessible to engineers

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 variation 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.