Positivity 13 (2009), 683–692
2009 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040683-10, published online February 6, 2009
The duality problem for the Class
of b-weakly compact operators
Belmesnaoui Aqzzouz, Aziz Elbour and Jawad Hmichane
Abstract. We establish necessary and suﬃcient conditions under which
b-weakly compact operators between Banach lattices have b-weakly compact
adjoints or operators with b-weakly compact adjoints are themselves b-weakly
compact. Also, we give some consequences.
Mathematics Subject Classiﬁcation (2000). 46A40, 46B40, 46B42.
Keywords. b-weakly compact operator, Order continuous norm, KB-space.
1. Introduction and notation
The space of b-weakly compact operators appeared for the ﬁrst time in the paper
. It is a class bigger than the class of weakly compact operators, but smaller than
the class of order weakly compact operators, which was introduced by Dodds in .
The deﬁnition is based on a new notion called b-order bounded subsets introduced
in . Contrary to weakly compact operators [2,9], the class of b-weakly compact
operators satisﬁes the domination problem i.e. if E and F are two Banach lattices,
and S and T are two operators from E into F such that 0 ≤ S ≤ T and T
is b-weakly compact, then S is b-weakly compact (, Corollary 2.9). But one
of the short comings of this class, is that it does not satisfy the duality property.
More precisely, there is a b-weakly compact operator whose adjoint is not b-weakly
compact and conversely, there is an operator which is not b-weakly compact while
its adjoint is one. In fact, the identity operator of the Banach lattice l
compact but its adjoint, which is the identity operator of the Banach lattice l
is not b-weakly compact. Conversely, the identity operator of the Banach lattice
, is not b-weakly compact but its adjoint, which is the identity operator of the
Banach lattice l
, is b-weakly compact.
The goal of this paper is to characterize the class of Banach lattices under
which the duality problem has a positive solution for the space of b-weakly compact
operators. In fact, after some preliminaries on this class of operators, we will
prove that if E and F are Banach lattices, then each b-weakly compact operator