  We give some characterizations of disjointly weakly compact sets in Banach
lattices, namely, those sets in whose solid hulls every disjoint sequence
converges weakly to zero. As an application, we prove that a bounded linear
operator from a Banach space to a Banach lattice is an almost $(L)$ limited
operator if and only if it is a disjointly weakly compact operator, indeed, an
operator which carries bounded sets to disjointly weakly compact ones. Some
results on weak precompactness and ($L$-, $M$-)weak compactness of disjointly
weakly compact operators are also obtained.
