Factorization Theorems for Multiplication Operators on Banach Function Spaces

摘要

Let X Y and Z be Banach function spaces over a measure space (Ω,Σ, μ). Consider the spaces of multiplication operators XY′ from X into the K?the dual Y′of Y, and the spaces X~Z and ZY′ defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space X~Z · Z~Y′ ? XY′that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey–Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class d_(p,Z)~? of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.