A Hill-Pick matrix criterion for the Lyapunov order

Abstract

The Lyapunov order appeared in the study of Nevanlinna-Pick interpolation for positive real odd functions with general (real) matrix points. For real or complex matrices A and B it is said that B Lyapunov dominates A ifH=H⁎,HA+A⁎H⩾0⇒HB+B⁎H⩾0. (In case A and B are real we usually restrict to real Hermitian matrices H, i.e., symmetric H.) Hence B Lyapunov dominates A if all Lyapunov solutions of A are also Lyapunov solutions of B. In this paper we restrict to the case that appears in the study of Nevanlinna-Pick interpolation, namely where B is in the bicommutant of A and where A is Lyapunov regular, meaning the eigenvalues λj of A satisfyλi+λ‾j≠0,i,j=1,…,n. In this case we provide a matrix criterion for Lyapunov dominance of A by B. The result relies on a class of ⁎-linear maps for which positivity and complete positivity coincide and a representation of ⁎-linear matrix maps going back to work of R.D. Hill. The matrix criterion asks that a certain matrix, which we call the Hill-Pick matrix, be positive semidefinite.