We present a close relationship between row, column and doubly stochastic operators and the majorization relation on a Banach space $\ell^p(I)$, 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 $\ell^p(I)$, $p\in[1,\infty)$, and consider relations between this concept and the majorization on $\ell^p(I)$ mentioned above. Moreover, we give conditions that generalized Kakutani's conjecture is true.