Zygmund's theorem and the boundary behavior of operator \(R\)-functions
type: Item
instance of: scholarly article
title: Zygmund's theorem and the boundary behavior of operator \(R\)-functions
zbMATH Open document ID: 0901.47010
DOI: 10.1007/BF02509510
author: S. N. Nakobo
publication date: 1997-06-25T00:00:00Z
review text: An important class of operator \(R\)-functions of the form \(T(\lambda)= V^{1/2}(L-\lambda I)^{-1} V^{1/2}\), \(\lambda\in\mathbb{C}_+\), is considered, where \(V\) is a compact nonnegative operator and \(L\) is a selfadjoint operator in a Hilbert space \(H\). The boundary behaviour of such \(R\)-functions is considered, and in particular the problem when the boundary values \(T(k+ i\cdot 0)\) belong to the Schatten class \({\mathfrak S}_1\) for almost all \(k\in R\). The author first considers the Zygmund condition \[ \sum_n s_n(V)\log^+(1/s_n(V))< \infty,\tag{1} \] which is equivalent to the condition \[ \sum_n s_n(V)\ln n< \infty,\tag{2} \] and proves that this condition does not guaranty that the boundary values \(T(k+ i\cdot 0)\) belong to \({\mathfrak S}_1\) for almost all \(k\in R\). Hence the operator analog of the scalar Zygmund theorem fails. Then he describes the Marcinkiewicz classes \({\mathfrak S}_\pi\) (the duals of the Lorentz classes) to which the boundary values \(T(k+i\cdot 0)\) of such \(R\)-functions can belong for almost all \(k\in R\).
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 1023828
zbMATH Keywords: operator analog scalar Zygmund theorem fails
zbMATH Keywords: operator \(R\)-functions
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: Schatten class
zbMATH Keywords: Zygmund condition
zbMATH Keywords: Marcinkiewicz classes
zbMATH Keywords: duals of the Lorentz classes
reviewed by: Aleksandar Torgašev
MaRDI profile type: Publication
cites work: Uniqueness theorems for operator-valued functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model
cites work: Description of contraction operators which are similar to unitary operators
cites work: Q3874458
full work available at URL: https://doi.org/10.1007/bf02509510
OpenAlex ID: W2059038899
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1023828