THE BISHOP-PHELPS-BOLLOBAS THEOREM AND ASPLUND OPERATORS

This paper deals with a strengthening of the Bishop-Phelps property for operators that in the literature is called the Bishop-Phelps-Bollobas property. Let X be a Banach space and L a locally compact Hausdorff space. We prove that if T : X -> C-0(L) is an Asplund operator and parallel to T(x(0))parallel to parallel to T parallel to for some parallel to x(0)parallel to = 1, then there is a norm-attaining Asplund operator S : X -> C-0(L) and parallel to u(0)parallel to = 1 with parallel to S(u(0))parallel to = parallel to S parallel to = parallel to T parallel to such that u(0) sic x(0) and S sic 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-Bollobas property for all L; (C) if L is scattered, the pair (X, C-0(L)) has the Bishop-Phelps-Bollobas property for all Banach spaces X.