A Hill-Pick matrix criteria for the Lyapunov order

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 $A$ if \begin{equation*} H=H^*,\quad HA+A^*H \geq 0 \quad \implies \quad HB+B^*H \geq 0. \end{equation*} (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 chapter 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 $\lambda_j$ of $A$ satisfy $$\lambda_i + \overline{\lambda}_j \ne 0, \quad i,j=1,\ldots,n.$$ In this case we provide a matrix criteria 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 criteria asks that a certain matrix, which we call the Hill-Pick matrix, be positive semidefinite.