2021/11/17 by S. ter Horst, ter Horst, Sanne, Alma van der Merwe +1
Computer Science · Mathematics · #15A39 #15B05 #15B48 #30E05 #46L07 #47A57 #47L07 #93D05 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Primary 93D30 #Secondary 15A04 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.08979
openalex publication_date 2021/11/17 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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 H=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 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 λj of A satisfy λi + λj ≠ 0, i,j=1,…,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.