Doubly (sub)stochastic operators on ℓp spaces

Abstract

A square matrix is said to be doubly stochastic if its elements are non-negative and all row sums and column sums are equal one. An important tool in the study of majorization for infinite dimensional spaces ℓp(I) are doubly stochastic operators. These operators are a generalization of doubly stochastic matrices. In this paper, we compare some properties of doubly stochastic operators in finite and infinite dimensions. We will see that if D:ℓp(I)→ℓp(I) is a doubly stochastic operator then ‖D‖≤1. Moreover, the existence of such an operator with ‖D‖<1 is equivalent to 1<p<∞ and I is an infinite set. We discuss some other properties of doubly stochastic operators such as compactness and closedness and also, provide relevant applications of these operators in the existence of solutions for some infinite linear equations and functional equations.