Criteria for the strong regularity of \(J\)-inner functions and \(\gamma\)-generating matrices.
type: Item
instance of: scholarly article
title: Criteria for the strong regularity of \(J\)-inner functions and \(\gamma\)-generating matrices.
zbMATH Open document ID: 1044.47012
DOI: 10.1016/S0022-247X(03)00067-2
author: Harry Dym
author: Damir Z. Arov
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-27T00:00:00Z
review text: Let \(J\) be an \(m\times m\) signature matrix \((J= J^*= J^{-1})\), and let \(\Omega_+\) be either the open unit disk or the open upper halfplane. An \(m\times m\)-matrix valued function \(U(\lambda)\) that is meromorphic in \(\Omega_+\) is called \(J\)-inner if \(U(\lambda)\) is \(J\)-unitary for a.e. point \(\lambda\) on the boundary of \(\Omega_+\) and \(J- J(\lambda)^* JU(\lambda)\) is positive semidefinite for every point \(\lambda\in \Omega_+\) of analyticity of \(U(\lambda)\). The class of right strongly regular \(J\)-inner functions was introduced in previous works of the authors in connection with their study of canonical integral and differential systems. For \(J= \left[ \begin{smallmatrix} I_p &0\\ 0 &-I_q \end{smallmatrix} \right]\) (here \(p+q= m\)), a \(J\)-inner function \(U(\lambda)= \left[ \begin{smallmatrix} w_{11}(\lambda) &w_{12}(\lambda)\\ w_{21}(\lambda) &w_{22}(\lambda) \end{smallmatrix} \right]\), where \(w_{11}(\lambda)\) is \(p\times p\) and \(w_{22}(\lambda)\) is \(q\times q\), is called right strongly regular if \[ \|(w_{11} \varepsilon+ w_{12}) (w_{21} \varepsilon+ w_{22})^{-1} \|_\infty< 1 \] for at least one \(p\times q\)-function \(\varepsilon\) which is analytic and contractive in \(\Omega_+\). Characterizations of right strongly regular \(J\)-inner functions in terms of the matricial Muckenhoupt condition were obtained in [\textit{D. A. Arov} and \textit{H. Dym}, J. Funct. Anal. 181, 227--299 (2001; Zbl 0980.47014)]. A positive semidefinite matrix function \(\Delta(\lambda)\) is said to satisfy the matrix Muckenhoupt condition if \[ \sup_I \|(A_I(\Delta))^{1/2} (A_I (\Delta^{-1}))^{1/2}\|< \infty, \] where \(A_I(\Delta)\) is the average of \(\Delta\) of a finite subinterval \(I\) of the boundary of \(\Omega_+\) having positive length. In the paper under review, different characterizations are obtained. One of the main results is as follows: A \(\left[ \begin{smallmatrix} I_p &0\\ 0 &-I_q \end{smallmatrix} \right]\)-inner function \…
reviewed by: Leiba Rodman
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 1917781
zbMATH Keywords: \(J\)-inner matrix valued functions
zbMATH Keywords: matricial Nehari problem
zbMATH Keywords: \({\gamma}\)-generating matrices
zbMATH Keywords: strong regularity
zbMATH Keywords: bitangential interpolation problems
zbMATH Keywords: matricial Muckenhoupt condition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1917781