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

TL;DR

The paper demonstrates that a fixed-operator version of the Bishop-Phelps-Bollobás property cannot generally hold for higher-dimensional Banach spaces, highlighting a fundamental limitation in operator approximation.

Contribution

It proves that the Bishop-Phelps-Bollobás property without changing the operator is only valid when one of the Banach spaces involved is one-dimensional.

Findings

Constructs sequences of operators with specific approximation properties.

Shows the impossibility of a fixed-operator Bishop-Phelps-Bollobás property in higher dimensions.

Highlights the dimensional restriction for the property to hold.

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.