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

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