Majorization and doubly stochastic operators

Abstract

We present a close relationship between row, column and doubly stochastic operators and the majorization relation on a Banach space ?p(I), where I is an arbitrary non-empty set and p ? [1,?]. Using majorization, we point out necessary and sufficient conditions that an operator D is doubly stochastic. Also, we prove that if P and P-1 are both doubly stochastic then P is a permutation. In the second part we extend the notion of majorization between doubly stochastic operators on ?p(I), p ? [1,?), and consider relations between this concept and the majorization on ?p(I) mentioned above. Moreover, we give conditions that generalized Kakutani?s conjecture is true.