Majorization and semidoubly stochastic operators on L 1 ( X ) $L^{1}(X)$ - Seyed Mahmoud Manjegani; Shirin Moein (Journal of Inequalities and Applications, 2023)
Abstract This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by S D ( L 1 ) $S\mathcal{D}(L^{1})$ ) on L 1 ( X ) $L^{1}(X)$ 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 ) $L^{1}(X)$ . 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 = { S f : S ∈ S D ( L 1 ) } $S_{f}=\{Sf : S\in S\mathcal{D}(L^{1})\}$ for a fixed element f ∈ L 1 ( X ) $f\in L^{1}(X)$ by proving the equiintegrability of S f $S_{f}$ . We present a full characterization of majorization on a σ-finite measure space X.