Minimal extensions in tensor product spaces
type: Item
instance of: scholarly article
title: Minimal extensions in tensor product spaces
zbMATH Open document ID: 0927.46015
author: Grzegorz Lewicki
published in: Journal of Approximation Theory
publication date: 1999-12-13T00:00:00Z
review text: Let \(X\) be any Banach space and let \(V\subseteq S\subseteq X\) be subspaces of \(X\) with \(\dim V<\infty\). For any operator \(M\in {\mathcal L}(S,V)\), the author defines the minimal extension constant by \[ \lambda_M(V, X)= \inf\{\| P\|: P\in{\mathcal L}(X, V), P|_S= M\}. \] For \(V= S\) and \(M= 1_V\) this generalizes the concept of a minimal projection constant \(\lambda(V,X)\). In the paper under review, the author proves as a main result the following nice formula: Let \(X\), \(Y\) be any separable Banach spaces and let \(V\subseteq S\subseteq X\) and \(W\subseteq Z\subseteq Y\) be any subspaces with \(\dim V\), \(\dim W<\infty\). Let \(\alpha\) be any reasonable and uniform crossnorm on \(X\otimes Y\). Then \[ \lambda_{M\otimes N}(V\otimes_\alpha W, X\otimes_\alpha Y)= \lambda_M(V, X)\lambda_N(W, Y) \] holds true for all \(M\in {\mathcal L}(S, V)\) and \(N\in{\mathcal L}(Z, W)\), respectively. The proof is essentially based on a quite recent characterization of minimal extensions of operators by certain integral operators given by \textit{B. L. Chalmers} and \textit{F. T. Metcalf} [J. Oper. Theory 32, 31-46 (1994; Zbl 0827.41016)]. In the final section, the main result is applied to function spaces and subspaces of polynomials of fixed degree.
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 47A58
zbMATH DE Number: 1278849
zbMATH Keywords: minimal extension constant
zbMATH Keywords: minimal projection constant
zbMATH Keywords: subspaces of polynomials of fixed degree
reviewed by: Heinz Junek
MaRDI profile type: Publication
full work available at URL: jath.1998.3246
OpenAlex ID: W1993573576
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DOI: 10.1006/JATH.1998.3246
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description: scientific article; zbMATH DE number 1278849