The Bishop-Phelps-Bollobás property for operators on \(\mathcal{C}(K)\)
type: Item
instance of: scholarly article
title: The Bishop-Phelps-Bollobás property for operators on \(\mathcal{C}(K)\)
zbMATH Open document ID: 1347.46005
DOI: 10.1215/17358787-3492875
author: María D. Acosta
published in: Banach Journal of Mathematical Analysis
publication date: 2016-04-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1405.6428
full work available at URL: https://projecteuclid.org/euclid.bjma/1458053864
review text: This paper starts with an overview of known results that lead to the Bishop-Phelps-Bollobás property for operators. Consider \(X\) and \(Y\), Banach spaces, and \(\mathcal{L}(X,Y)\), the space of bounded operators from \(X\) into \(Y\). The unit sphere in \(X\) is denoted by \(S_X\) and the set of all norm 1 operators in \(\mathcal{L}(X,Y)\) is denoted by \(\mathcal{S}(X,Y)\). The pair \((X,Y)\) has the BPBp if for every \(\epsilon >0,\) there exists \(\eta (\epsilon)>0\) such that for every \(T\in \mathcal{S}(X,Y),\) and if \(x_0 \in S_X\) is such that \(\|Tx_0\|>1-\eta (\epsilon)\), then there exists a \(u_0 \in S_X\) and \(S\in \mathcal{S}(X,Y)\) with \(\|S(u_0)\|=1\), \(\|T-S\|<\epsilon\) and \(\|u_0-x_0\|<\epsilon\). In the introduction, the author also gives a list of pairs of Banach spaces that satisfy the Bishop-Phelps-Bollobás property for operators which motivates well the main theorem in the paper. Section 2 starts with the definition of \(\mathbb{C}\)-modulus of convexity of a complex Banach space \(Y\), given \(\epsilon>0\), \[ \delta(\epsilon) =\inf \{ \sup \{\|x+\lambda \epsilon y\|-1: \, \lambda \in \mathbb{C}, \, |\lambda|=1\}: x,y \in S_X\}. \] The Banach space \(Y\) is \(\mathbb{C}\)-uniformly convex if \(\delta (\epsilon)>0\) for every \(\epsilon >0\). This section is devoted to the proof of the following theorem. Theorem (cf. Theorem 2.4). The pair \((C_0(L), Y)\) satisfies the Bishop-Phelps-Bollobás property for operators for any locally compact Hausdorff topological space \(L\) and any \(\mathbb{C}\)-uniformly convex space \(Y\). Moreover, the function \(\eta\) appearing in the definition of BPBp depends only on the \(\mathbb{C}\)-modulus of convexity of \(Y\). Since the complex spaces \(L_p (\mu)\) (\(1\leq p<\infty\)) are \(\mathbb{C}\)-uniformly convex the author obtains the following result. Corollary (cf. Corollary 2.5). In the complex case, the pair \((C_0(L), L_p (\mu))\) (\(1\leq p<\infty\)) satisfies the Bishop-Phelps-Bollobás propert…
reviewed by: Fernanda Botelho
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47A99
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 6575513
zbMATH Keywords: Banach space
zbMATH Keywords: norm-attaining operators
zbMATH Keywords: denseness of norm-attaining operators
zbMATH Keywords: Bishop-Phelps-Bollobás property
MaRDI profile type: Publication
OpenAlex ID: W1850159273
arXiv ID: 1405.6428
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description: scientific article; zbMATH DE number 6575513