Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series

Abstract

In this paper, we characterize the space of multiplication operators from an L ^p-space into a space L ¹(m) of integrable functions with respect to a vector measure m, as the subspace $L^1_{{\rm p},\mu}({\bf m})$ defined by the functions that have finite p-semivariation. We prove several results concerning the Banach lattice structure of such spaces. We obtain positive results—for instance, they are always complete, and we provide counterexamples to prove that other properties are not satisfied—for example, simple functions are not in general dense. We study the operators that factorize through $L^1_{{\rm p},\mu}({\bf m})$, and we prove an optimal domain theorem for such operators. We use our characterization to generalize the Bennet–Maurey–Nahoum Theorem on decomposition of functions that define an unconditionally convergent series in L ¹[0,1] to the case of 2-concave Banach function spaces.