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

Abstract

We note that the well-known result of Von Neumann [29] is not valid for all doubly substochastic operators on discrete Lebesgue spaces ℓ p (I), p ∈ [1, ∞). This fact lead us to distinguish two classes of these operators. Precisely, the class of proper 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 ≺ s on the positive cone ℓ p (I) + , when p ∈ [1, ∞), is introduced by proper substochastic operator and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization ≺ s on ℓ 1 (I) + and on ℓ p (I) + , when I is an infinite set, are presented.