 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 139, Number 10, October 2011, Pages 35533560
S 00029939(2011)10755X
Article electronically published on February 10, 2011
THE BISHOPPHELPSBOLLOB´AS THEOREM
AND ASPLUND OPERATORS
R. M. ARON, B. CASCALES, AND O. KOZHUSHKINA
(Communicated by Nigel J. Kalton)
Dedicated to the memory of Nigel J. Kalton
Abstract. This paper deals with a strengthening of the BishopPhelps prop
erty for operators that in the literature is called the BishopPhelpsBollob´as
property. Let X be a Banach space and L a locally compact Hausdorff space.
We prove that if T : X C0(L) is an Asplund operator and T(x0) T
for some x0 = 1, then there is a normattaining Asplund operator S : X
C0(L) and u0 = 1 with S(u0) = S = T such that u0 x0 and S 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 = c0), the
pair (X, C0(L)) has the BishopPhelpsBollob´as property for all L; (C) if L is
scattered, the pair (X, C0(L)) has the BishopPhelpsBollob´as property for all
