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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 139, Number 10, October 2011, Pages 3553­3560
S 0002-9939(2011)10755-X
Article electronically published on February 10, 2011
THE BISHOP-PHELPS-BOLLOB´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 Bishop-Phelps prop-
erty for operators that in the literature is called the Bishop-Phelps-Bollob´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 norm-attaining 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 Bishop-Phelps-Bollob´as property for all L; (C) if L is
scattered, the pair (X, C0(L)) has the Bishop-Phelps-Bollob´as property for all

  

Source: Aron, Richard M. - Department of Mathematics, Kent State University

 

Collections: Mathematics