In this paper, we introduce the quasidoubly stochastic operator, which is between doubly stochastic operators and column stochastic operators, so as to apply to characterized operator S on l1 such that Sf is majorized by f for every f ∈ l1. We present some classes of majorization preservers on l1 under quasi doubly stochastic operators. Moreover, as an application of our result in quantum physics, the convertibility of pure states of a composite system by local operations and classical communication has been considered.

We consider positive, integral-preserving linear operators acting on L 1 space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering matrix majorization in L 1 ). We collect a number of results for vector-valued functions on L 1 , simplifying some proofs found in the literature. In particular, matrix majorization and multivariate majorization are related in R n . In R, these are also equivalent to convex function inequalities.2010 Mathematics Subject Classification. 15B51, 26B25, 26D15, 47B65 .

No abstract

This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by $S\mathcal{D}(L^{1})$ S 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 $S_{f}=\{Sf : S\in S\mathcal{D}(L^{1})\}$ S f = { S f : S ∈ S D ( L 1 ) } for a fixed element $f\in L^{1}(X)$ f ∈ 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.