Some extensions of Loewner's theory of monotone operator functions.
type: Item
instance of: scholarly article
title: Some extensions of Loewner's theory of monotone operator functions.
zbMATH Open document ID: 1075.47008
author: Vladimir Bolotnikov
author: Daniel Alpay
author: A. Dijksma
author: James Rovnyak
published in: Journal of Functional Analysis
publication date: 2002-08-04T00:00:00Z
full work available at URL: 1 s2.0 S002212360193856X main.pdf
review text: There are many results devoted to the class of real-valued differentiable functions \(f\) on \((a;b)\subset \mathbb R^1\) such that \(f(A)\leq f(B)\) whenever \(A\) and \(B\) (\(A\leq B\)) are selfadjoint matrices with eigenvalues lying in \((a;b)\). This class coincides with the set of all real-valued differentiable functions \(f\) on \((a;b)\) for which the Hermitian kernel \(K_f(x;y)\geq 0\) \(\forall x,y \in (a;b)\). Moreover, the Hermitian kernel \(K_f\) is nonnegative iff a function \(f\) is the restriction of a Nevanlinna function which has an analytic continuation across \((a;b)\). The paper under review considers a matrix-valued function \(F\) instead of the scalar-valued function \(f\). Several extensions of Loewner's theory of monotone operator functions are obtained. Some theorems on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class are proved. Further generalizations are connected with the monotone functions of an operator argument. The authors introduce and study, by analogy with classical Nevanlinna theory, a notion of \(k\)-monotonicity for such operator functions. It should be noted that there are corresponding results for Stieltjes functions.
reviewed by: Igor M. Mitelman
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47A63
zbMATH DE Number: 1747014
MaRDI profile type: Publication
OpenAlex ID: W2045657553
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DOI: 10.1006/JFAN.2001.3856
description: scientific article; zbMATH DE number 1747014