A criterion of contractivity for four by four matrices and applications to model operators
Axel Renard · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
We establish an explicit criterion for determining whether a $4 \\times 4$ upper-triangularmatrix is a contraction with respect to the Euclidean operator norm. Following this,we present a characterization of the matrix representation of a finite-dimensionalmodel operator acting on the space $H^2\\left(\\mathbb{D}\\right) ⊖ uH^2\\left(\\mathbb{D}\\right)$. Here, $u$ denotes a finite Blaschke product of degree $n$. We prove that the model matrix, when written inthe Takenaka-Malmquist-Walsh basis, is the only $n \\times n$ contractive matrix with prescribed main and upper diagonals.