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 these operators. Precisely, class increasable $\ell^p(I)$ isolated with property an analogue in this true. The submajorization relation $\prec_s$ positive cone $\ell^p(I)^+$, when $p\in[1,\infty)$, introduced by operator and it provided may be considered as a partial order. Two different shapes linear preservers $\ell^1(I)^+$ $I$ infinite set, are presented.