Maurey–Rosenthal factorization of positive operators and convexity

We show that a Banach lattice X is r-convex, 1 < r <∞, if and only if all positive operators T on X with values in some r-concave Köthe function spaces F(ν) (over measure spaces (Ω, ν)) factorize strongly through Lr (ν) (i.e., T =MgR, where R is an operator from X to Lr (ν) and Mg a multiplication operator on Lr (ν) with values in F). This characterization of r-convexity motivates a Maurey–Rosenthal type factorization theory for positive operators acting between vector valued Köthe function spaces.  2004 Elsevier Inc. All rights reserved