The Bishop-Phelps-Bollobás property and absolute sums (Q2424116)

A pair of Banach spaces $(X,Y)$ is said to have the Bishop-Phelps-Bollobás property (BPBp) if, for every $\varepsilon>0$, there is some $\eta>0$ such that the following holds: for every norm-one operator $T:X \rightarrow Y$ and every $x_0\in X$ such that $\|x_0\|=1$ and $\|Tx_0\|>1-\eta$, there exist a norm-one operator $S:X \rightarrow Y$ and an element $x_1\in X$ such that $\|x_1\|=1=\|Sx_1\|$, $\|x_0-x_1\|<\varepsilon$ and $\|T-S\|<\varepsilon$. Roughly speaking, this means that the famous Bishop-Phelps-Bollobás theorem for norm-attaining functionals also holds for operators from $X$ to $Y$. The authors study the question under which conditions the BPBp passes from the pair $(X,Y)$ to pairs of the form $(X_1,Y)$ or $(X,Y_1)$, where $X_1$ is an absolute summand of $X$ and $Y_1$ is an absolute summand of $Y$. Analogous questions are also studied for the BPBp for compact operators and for the density of norm-attaining operators. The same questions are also treated for the case that the operator-norm is replaced by the numerical radius.