Linear maps preserving or strongly preserving majorization on matrices

For A,B∈M nm, A,B∈Mnm, we say that A A is left matrix majorized (resp. left matrix submajorized) by B B and write A≺ ℓ B A≺ℓB (resp. A≺ ℓs B A≺ℓsB), if A=RB A=RB for some n×n n×n row stochastic (resp. row substochastic) matrix R. R. Moreover, we define the relation ∼ ℓs ∼ℓs on M nm Mnm as follows: A∼ ℓs B A∼ℓsB if A≺ ℓs B≺ ℓs A. A≺ℓsB≺ℓsA. This paper characterizes all linear preservers and all linear strong preservers of ≺ ℓs ≺ℓs and ∼ ℓs ∼ℓs from M nm Mnm to M nm Mnm.