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

Abstract

Given two real Banach spacesXandYwith dimensions greater than one, it is shown that there is a sequence{Tn}n is an element of Nof norm attaining norm-one operators fromXtoYand a pointx0 is an element of Xwith parallel to x0 parallel to=1, such that parallel to Tn(x0)parallel to 1 but infn is an element of N{dist(x0,{x is an element of X:parallel to Tn(x)parallel to=parallel to x parallel to=1})}>0.This shows that a version of the Bishop-Phelps-Bollobas property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.