Optimal Extensions for pth Power Factorable Operators

Let X(μ) be a function space related to a measure space (Ω,Σ, μ) with χΩ ∈ X(μ) and let T : X(μ) → E be a Banach spacevalued operator. It is known that if T is pth power factorable then the largest function space to which T can be extended preserving pth power factorability is given by the space Lp(mT) of p-integrable functions with respect to mT, where mT : Σ → E is the vector measure associated to T via mT (A) = T(χA). In this paper, we extend this result by removing the restriction χΩ ∈ X(μ). In this general case, by considering mT defined on a certain δ-ring, we show that the optimal domain for T is the space Lp(mT )∩L1(mT ). We apply the obtained results to the particular case when T is a map between sequence spaces defined by an infinite matrix.