Submajorization on lp(I)+ determined by increasable doubly substochastic operators and its linear preservers

We note that the well-known result of von Neumann (Contrib Theory Games 2:5-12, 1953) is not valid for all doubly substochastic operators on discrete Lebesgue spaces l(p)(I), p is an element of [1, infinity). This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on l(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 (sic)(s) on the positive cone l(p)(I)(+), when p is an element of [1, infinity), is introduced by increasable substochastic operators and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization (sic)(s) on l(1)(I)(+) and on l(p)(I)(+), when I is an infinite set, are presented.