CERTAIN CLASSES OF SETS IN THE DUAL OF A BANACH LATTICE AND SPACES OF OPERATORS

Abstract

Motivated by the definition of L-sets and using positively limited
weakly p-summable sequences in a Banach lattice, the notion of L-positively
limited sets of order p (1 ≤ p ≤ ∞) is introduced in the dual of a Banach
lattice. The connection between them with relatively weakly compact sets and
weak∗-sequentially compact sets is also discussed. Moreover, using positively
limited p-convergent operators, some operator characterizations of a Banach
lattice with the L-positively limited property of order p are obtained. Finally,
by almost positively limited p-convergent operators which will be defined, an
operator characterization of order continuity of the norm is provided.