Majorization and semidoubly stochastic operators on L^{1}(X)

Abstract

This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by Smathcal{D}(L^{1})) on L^{1}(X) when X is a σ-finite measure space. We answer Mirsky’s question and characterize the majorization by means of semidoubly stochastic maps on L^{1}(X). We prove some results on semidoubly stochastic operators such as a strong relation of semidoubly stochastic operators and integral stochastic operators and relatively weak compactness of S_{f}={Sf : Sin Smathcal{D}(L^{1})} for a fixed element fin L^{1}(X) by proving the equiintegrability of S_{f}. We present a full characterization of majorization on a σ-finite measure space X.