GUT-majorization and its linear preservers

Abstract

Dedicated to the memory of Batool Bagheri, the founder of mathematics house of Kerman.Abstract. Let Mn,m be the set of all n × m real matrices. An n × m matrix R = [r ij ] is g-row stochastic if m k=1 r ik is equal to 1 for all i (1 ≤ i ≤ n). Let X, Y ∈ Mn,m. The matrix X is said to be gut-majorized by Y (denoted by X ≺gut Y ), if there exists an n × n upper triangular g-row stochastic matrix R such that X = RY . Recall that a linear function T : Mn,m → Mn,m preserves (or strongly preserves) a relation ∼, if T X ∼ T Y whenever X ∼ Y (or T X ∼ T Y if and only if X ∼ Y ). The present papers establishes some facts about ≺gut on Mn,m, and characterizes the structure of all (strong) linear preservers of ≺gut on R n and strong linear preservers of ≺gut on Mn,m.

On linear preservers of two-sided gut-majorization on $ M_{n,m}$

“…Let X, Y ∈ M n,m such that X ∼ gut Y . [1], Theorem 1.3 ensures that T strongly preserves ≺ gut . So X ∼ gut Y if and only if X ≺ gut Y ≺ gut X if and only if T X ≺ gut T Y ≺ gut T X if and only if T X ∼ gut T Y .…”

“…The (strong) linear preservers and strong preservers of ≺ gut on R n and M n,m are fully characterized in [1]. For more information about linear preservers of majorization we refer the reader to [2]- [10].…”