Majorization and doubly stochastic operators

We present a close relationship between row, column and doubly stochastic operators and the majorization relation on a Banach space lp(I), where I is an arbitrary non-empty set and p \in [1,\infty]. 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 lp(I), p \in [1,\infty), and consider relations between this concept and the majorization on lp(I) mentioned above. Moreover, we give conditions that generalized Kakutani’s conjecture is true.