The Bishop-Phelps-Bollobás theorem and Asplund operators

Abstract:

This paper deals with a strengthening of the Bishop-Phelps property for operators that in the literature is called the Bishop-Phelps-Bollobás property. Let $X$ be a Banach space and $L$ a locally compact Hausdorff space. We prove that if $T:X\to C_0(L)$ is an Asplund operator and $\|T(x_0)\|\approxeq \|T\|$ for some $\|x_0\|=1$, then there is a norm-attaining Asplund operator $S:X\to C_0(L)$ and $\|u_0\|=1$ with $\|S(u_0)\|= \|S\|=\|T\|$ such that $u_0\approxeq x_0$ and $S\approxeq T$. As particular cases we obtain: (A) if $T$ is weakly compact, then $S$ can also be taken to be weakly compact; (B) if $X$ is Asplund (for instance, $X=c_0$), the pair $(X,C_0(L))$ has the Bishop-Phelps-Bollobás property for all $L$; (C) if $L$ is scattered, the pair $(X,C_0(L))$ has the Bishop-Phelps-Bollobás property for all Banach spaces $X$.