Positivity 10 (2006), 51–63
2006 Birkh¨auser Verlag Basel/Switzerland
The Alternative Dunford–Pettis Property
for Subspaces of the Compact Operators
Mar´ıa D. Acosta and Antonio M. Peralta
Abstract. A Banach space X has the alternative Dunford–Pettis property if
for every weakly convergent sequences (x
) → x in X and (x
) → 0inX
= x =1wehave(x
)) → 0. We get a characterization of
certain operator spaces having the alternative Dunford–Pettis property. As a
consequence of this result, if H is a Hilbert space we show that a closed sub-
space M of the compact operators on H has the alternative Dunford–Pettis
property if, and only if, for any h ∈ H, the evaluation operators from M
to H given by S → Sh, S → S
h are DP1 operators, that is, they apply
weakly convergent sequences in the unit sphere whose limits are also in the
unit sphere into norm convergent sequences. We also prove a characterization
of certain closed subalgebras of K(H) having the alternative Dunford-Pettis
property by assuming that the multiplication operators are DP1.
Keywords. The Dunford-Pettis property, the alternative Dunford-Pettis Prop-
erty, compact operators.
A Banach space X has the Dunford-Pettis property (DP in the sequel) if for any
Banach space Y , every weakly compact operator from X to Y is completely con-
tinuous, that is, it maps weakly compact subsets of X onto norm compact subsets
of Y . The DP was introduced by Grothendieck who also showed that a Banach
space X has the DP if, and only if, for every weakly null sequences (x
we have x
) → 0. Since its introduction by Grothendieck, the DP
has had an important development. We refer to Diestel  as an excellent survey
on the DP and to Bourgain , Bunce , Chu-Iochum  and Talagrand  and
Chum-Mellon  for more recent results.
Recently, S. Brown and A.
Ulger (see Brown  and
Ulger ) have stud-
ied the DP for subspaces of the compact operators on an arbitrary Hilbert space.
Indeed, if M is a closed subspace of the compact operators on a Hilbert space H,