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arXiv · 1810.00684

There is no operatorwise version of the Bishop-Phelps-Bollob\'as property

Sheldon Dantas·Vladimir Kadets·Sun Kwang Kim·Han Ju Lee·Miguel Martín
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Abstract

Given two real Banach spaces $X$ and $Y$ with dimensions greater than one, it is shown that there is a sequence $\{T_n\}_{n\in \mathbb{N}}$ of norm attaining norm-one operators from $X$ to $Y$ and a point $x_0\in X$ with $\|x_0\|=1$, such that $\|T_n(x_0)\|\longrightarrow 1$ but $\inf_{n \in \mathbb{N}} \{\mbox{dist} (x_0,\,\{x\in X: \|T_n(x)\|=\|x\|=1\})\} >0.$ This shows that a version of the Bishop-Phelps-Bollob\'as property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.

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Sheldon Dantas, Vladimir Kadets, Sun Kwang Kim, Han Ju Lee, Miguel Martín. 2018-10-01. There is no operatorwise version of the Bishop-Phelps-Bollob\'as property. https://doi.org/10.1080/03081087.2018.1560388

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