p-th Power Factorable Operators

Abstract

Let X(μ) be a q-B.f.s. with σ-o.c. norm and E be a Banach space. A continuous linear operator T : X(μ ) → E is p-th power factorable (cf. Definition 5.1 below), for 1 ≤ p ≺ ∞, if there exists $T_{[p]} \in \mathcal{L}(X(\mu )_{[p]} ,E)$ which coincides with T on X(μ) ⊆ X(μ)[_p]. There is no a priori reason to suspect any connection between the p-th power factorability of T and its associated E-valued vector measure m_T : A → T( X_A). The aim of this chapter is to convince the reader that such a connection does indeed exist and has some far-reaching consequences. Henceforth, assume that T is also μ-determined.