The Cohen-Lewkowicz approach to Nevanlinna- Pick interpolation

Abstract

The class of positive real odd functions, both scalar and matrix functions, are studied. Our point of departure is a comparison between the scalar, one port, case and the multi-port setting of functions in this class. We conclude our study by providing conditions for the existence of a scalar function f in this class such that fpAq = B, where A and B are given real square matrices with additional constraints. This matrix-valued interpolation problem, known as the Cohen-Lewkowicz interpolation problem, requires B to Lyapunov dominate A. One can express this condition by making use of the Lyapunov operator, which is a *-linear matrix map, i.e., it perserves adjoints. We consider *-linear matrix maps in general and expand the study of Hill representations of a *-linear matrix map in various directions, one being we determine the associated Hill matrix explicitly. This study leads us to classes of *-linear matrix maps for which positivity and complete positivity coincide. These classes of *-linear matrix maps are determined by structural conditions on a matrix associated with the linear map. Back in the setting of the Cohen-Lewkowicz interpolation problem, we provide a matrix criteria for Lyapunov dominance. This result relies on a class of *-linear maps for which positivity and complete positivity coincide, as well as the Hill representation of a *-linear map. The matrix criteria asks that a certain matrix, which we call the Hill-Pick matrix, be positive semidefinite, a matrix criteria similar to the Pick matrix criteria found in other interpolation settings. Finally, we make use of all our foregoing work to show the existence of a function f that solves the Cohen-Lekowicz interpolation problem.