The Bishop-Phelps-Bollobás theorem for \(\mathcal L(L_1 (\mu), L_\infty [0,1])\)
The Bishop-Phelps-Bollobás property for operators was recently introduced as an extension to operators of the quantitative version of the Bishop-Phelps theorem due to Bollobás. The paper under review is devoted to prove that the pair \((L_1(\mu),L_\infty[0, 1])\) has this property for every \(\sigma\)-finite measure \(\mu\). That is, given \(\varepsilon > 0\) it is shown that there are \(\eta(\varepsilon) > 0\) and \(\beta(\varepsilon) > 0\) with \(\lim_{\varepsilon\rightarrow0} \beta(\varepsilon) = 0\) such that for all norm one operators \(T \in L(L_1(\mu),L_\infty[0, 1])\), if \(x_0 \in S_{L_1(\mu)}\) is such that \(\|Tx_0\| > 1-\eta(\varepsilon)\), then there exist \(u_0 \in S_{L_1(\mu)}\) and a norm one operator \(S \in L(L_1(\mu),L_\infty[0, 1])\) that satisfy the following conditions: \[ \|Su_0\| = 1,\quad \|u_0-x_0\| < \beta(\varepsilon), \quad \text{and} \quad \|S-T\| < \varepsilon. \] This implies that the set of all norm-attaining operators is dense in \(L(L_1(\mu),L_\infty[0, 1])\) in a uniform way. So the above result is a strenghning of the main theorem in [\textit{C. Finet} and \textit{R Payá}, Isr. J. Math 108, 139--143 (1998; Zbl 0929.47037)].
scientific article; zbMATH DE number 5941219
scientific article