Linear Functions Preserving Sut-Majorization on R^n

Suppose M_n is the vector space of all n-by-n real matrices, and let R^n be the set of all n-by-1 real vectors. A matrix R ∈ M_n is said to be row substochastic if it has nonnegative entries and each row sum is at most 1. For x, y ∈ Rn, it is said that x is sut-majorized by y (denoted by x ≺sut y) if there exists an n-by-n upper triangular row substochastic matrix R such that x = Ry. In this note, we characterize the linear functions T : R^n → R^n preserving (resp. strongly preserving) ≺sut with additional condition T e1≠ 0 (resp. no additional conditions).