2012/08/10 by Anatolii Grinshpan, Grinshpan, Anatolii, Dmitry S. Kaliuzhnyi-Verbovetskyi +3
Mathematics · #15A15 #47A13 #47A20 #47A48 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:15A15 #msc:47A13 #msc:47A20 #msc:47A48
paper · pdf · doi:10.48550/arxiv.1208.2288
arxiv created 2012/08/10 · openalex publication_date 2012/08/10 · arxiv updated 2012/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For every multivariable polynomial p, with p(0)=1, we construct a determinantal representation p=det (I - K Z), where Z is a diagonal matrix with coordinate variables on the diagonal and K is a complex square matrix. Such a representation is equivalent to the existence of K whose principal minors satisfy certain linear relations. When norm constraints on K are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial q, q(0)=0, satisfies the von Neumann inequality, then 1-q admits a determinantal representation with K a contraction. On the other hand, every determinantal representation with a contractive K gives rise to a rational inner function in the Schur--Agler class.