Weak supermajorization and families as doubly superstochastic operators on ℓp(I)

We present necessary and sufficient conditions that a family A={aij:i,j∈I} of real numbers may be considered as a bounded linear operator on Banach spaces ℓ1(I) and ℓ∞(I), where I is an arbitrary non-empty set. Moreover, we get that these conditions are sufficient for a family to be a bounded linear operator on ℓp(I), for each p∈[1,∞]. Within this class of operators, the notion of doubly superstochastic operator is introduced as an extension of the doubly superstochastic matrix, and some of its essentially properties are presented. In the second part, we extend the notion of weak supermajorization relation on the Banach space ℓp(I) using doubly superstochastic operators, and present close relationship between this relation and superstochastic operators as generalisation well-known results in the theory of majorization. Among others, for two functions f,g∈ℓ1(I)+ we show that relations f≺wsg and g≺wsf hold if and only if there exist a permutation P such that g=Pf.