There is no operatorwise version of the Bishop-Phelps-Bollobás property

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-Bollobás property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.

The content of this paper overlaps with the old version of arXiv:1709.00032 (arXiv:1709.00032v1, submitted on 31 Aug 2017). Nevertheless, there is no intersection between the present version and the updated one of arXiv:1709.00032 (arXiv:1709.00032v2, submitted on 26 Sep 2018)