Submajorization on $\ell^p(I)^+$ determined by increasable doubly substochastic operators and its linear preservers

We note that the well-known result of Von Neumann \cite{von} is not valid for all doubly substochastic operators on discrete Lebesgue spaces $\ell^p(I)$, $p\in[1,\infty)$. This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on $\ell^p(I)$ is isolated with the property that an analogue of the Von Neumann result on operators in this class is true. The submajorization relation $\prec_s$ on the positive cone $\ell^p(I)^+$, when $p\in[1,\infty)$, is introduced by increasable substochastic operator and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization $\prec_s$ on $\ell^1(I)^+$ and on $\ell^p(I)^+$, when $I$ is an infinite set, are presented.