Subcouples of codimension one and interpolation of operators that almost agree
type: Item
instance of: scholarly article
title: Subcouples of codimension one and interpolation of operators that almost agree
zbMATH Open document ID: 1064.46066
author: Peter Sunehag
published in: Journal of Approximation Theory
publication date: 2004-10-14T00:00:00Z
review text: Let \(\bar X\) and \(\bar Y\) be two compatible couples of Banach spaces and let \(T_0: X_0\rightarrow Y_0\) and \(T_1: X_1\rightarrow Y_1\) be bounded linear operators. Let us assume that \(\Gamma\in (\Delta(\overline X))'\) and that \(T_0\) and \(T_1\) agree on \(\Delta(\bar X)\cap \ker \Gamma\), then, in general, we cannot interpolate to obtain that \(T_j\) extends to a bounded operator \(T:J_{\theta, p}(\overline X)\rightarrow J_{\theta, p}(\overline Y)\) where \(J_{\theta, p}\) is the classical \(J\)-method of interpolation. However, in this paper, the author defines a space \(J_{\theta, p; \Gamma}(\overline X)\) in such a way that \(T:J_{\theta, p; \Gamma}(\overline X)\rightarrow J_{\theta, p}(\overline Y)\) is bounded and he studies this new space determining, among other things, when \(J_{\theta, p; \Gamma}(\overline X)= J_{\theta, p}(\overline X)\) or \(J_{\theta, p; \Gamma}(\overline X)\) is a closed subspace of \(J_{\theta, p}(\overline X)\). He also defines the space \(J_{E; \Gamma}(\overline X)\) for the general real method of interpolation.
Mathematics Subject Classification ID: 46M35
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 2107877
zbMATH Keywords: interpolation theory
zbMATH Keywords: Banach space
zbMATH Keywords: Banach couple
zbMATH Keywords: subspace
zbMATH Keywords: subcouple
zbMATH Keywords: quotient couple
reviewed by: María J. Carro
MaRDI profile type: Publication
full work available at URL: j.jat.2004.06.009
OpenAlex ID: W2056399620
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DOI: 10.1016/J.JAT.2004.06.009
description: scientific article; zbMATH DE number 2107877