Interpolation of compact operators by the complex method and equicontinuity
type: Item
instance of: scholarly article
title: Interpolation of compact operators by the complex method and equicontinuity
zbMATH Open document ID: 0985.46007
author: Tomas P. Schonbek
published in: Indiana University Mathematics Journal
publication date: 2001-04-23T00:00:00Z
Mathematics Subject Classification ID: 46B70
Mathematics Subject Classification ID: 46M35
zbMATH DE Number: 1590126
zbMATH Keywords: interpolation by the complex method
zbMATH Keywords: compactness operators
zbMATH Keywords: equicontinuity
MaRDI profile type: Publication
full work available at URL: iumj.2000.49.1738
OpenAlex ID: W1965753968
DOI: 10.1512/IUMJ.2000.49.1738
review text: The author investigates the relation between interpolation by the complex method of compactness operators and equicontinuity. Given a Banach couple \((A_0,A_1)\), let \({\mathcal F}(A_0, A_1)\) be the Banach space of analytic functions used in the definition of the complex method. A bounded set \(E\subset{\mathcal F}(A_0, A_1)\) is said to be \(\theta\)-equicontinuous if, for every \(\varepsilon> 0\), there is \(\delta> 0\) such that \(\|f(\theta+ it)- f(\theta+ it')\|_{[\theta]}< \varepsilon\) for all \(f\in E\), whenever \(t,t'\in R\) and \(|t-t'|< \delta\).NEWLINENEWLINENEWLINEGiven two Banach couples \((A_0, A_1)\) and \((B_0, B_1)\), the author shows that if \(T: A_0\to B_0\) is compact, then compactness of \(T:(A_0, A_1)_{[\theta]}\to (B_0, B_1)_{[\theta]}\) is equivalent to \(\theta\)-equicontinuity of \(T(B_{F(A_0, A_1)})\). Where \(B_{F(A_0, A_1)}\) is the unit ball of \({\mathcal F}(A_0, A_1)\).
reviewed by: Joaquim Martin
description: scientific article