Abstract
For x,y∈ℝn, we say x is weakly submajorized (weakly supermajorized) by y, and write x≺ωy (x≺ωy), if ∑1kx[i]≤∑1ky[i], k=1,2,…,n (∑1kx(i)≥∑1ky(i), k=1,2,…,n), where x[i] (x(i)) denotes the ith component of the vector x↓ (x↑) whose components are a decreasing (increasing) rearrangment of the components of x. We characterize the linear maps that preserve (strongly preserve) one of the majorizations ≺ω or ≺ω.