Weak majorization and doubly substochastic operators on ℓp(I)ℓp(I)

We extend the notion of weak majorization on the Banach space ℓp(I)ℓp(I), where I   is a non-empty set and p∈[1,∞)p∈[1,∞), using doubly substochastic operators on ℓp(I)ℓp(I). The close relationship between weak majorization and these operators are presented. Among others, we show that f≺wgf≺wg and g≺wfg≺wf, iff there is a partial permutation P   such that g=Pfg=Pf. Using the specific type of positive increasing convex functions, a necessary and sufficient condition for f≺wgf≺wg and g≺wfg≺wf is given.