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-Bollob\'as property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.