Jet functions having indefinite Carathéodory-Pick matrices
type: Item
instance of: scholarly article
title: Jet functions having indefinite Carathéodory-Pick matrices
zbMATH Open document ID: 1064.30036
author: Alexander Kheifets
author: Leiba Rodman
author: Vladimir Bolotnikov
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: Given a nonnegative integer \(\kappa\) and a nonempty subset \(\Omega\) of the open unit disk, the class \(\mathcal{P}_\kappa(\Omega)\) consists of complex valued functions \(f\) with the domain of definition \(\Omega\) such that all Pick matrices \[ P(f;z_1,\ldots,z_n):= \left[ \frac{1-f(z_i)f^*(z_j)}{1-z_iz^*_j}\right]_{i,j=1}^n, \qquad z_1,\ldots,z_n\in\Omega \] have at most \(\kappa\) negative eigenvalues and at least one such matrix has exactly \(\kappa\) negative eigenvalues. It was a matter of preceding papers by the authors to characterize the functions from \(\mathcal{P}_\kappa(\Omega)\) in term of their ``meromorphic'' extension. In the present paper a class of functions \(f\) called jet valued functions such that \(f\) belongs to \(\mathbb{C}^{m(z)}\), where \(0\leq m\leq\infty\) is an integer valued function, and all Carathéodory-Pick matrices enjoy the same property as the aforementioned one, is introduced and characterized in several ways. Extension and interpolation results are proved for functions in the class.
Mathematics Subject Classification ID: 30E99
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 2083516
zbMATH Keywords: Jet functions
zbMATH Keywords: generalized Schur functions
zbMATH Keywords: Carathéodory-Pick matrices
zbMATH Keywords: negative squares
zbMATH Keywords: interpolation
zbMATH Keywords: pseudomultipliers
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: j.laa.2003.09.006
OpenAlex ID: W2086662424
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DOI: 10.1016/J.LAA.2003.09.006
description: scientific article; zbMATH DE number 2083516