Linear maps which preserve or strongly preserve weak majorization

For x, y is an element of R-n we say x is weakly submajorized (weakly supermajorized) by y, and write x <(omega)y (x < (omega)y), if Sigma(k)(1)x[i] <= Sigma(k)(1) y[i], k = 1, 2,..., n (Sigma(k)(1)x( i) >= Sigma(k)(1) y((i)), k = 1, 2,..., n), where x[i] (x((i))) denotes the ith component of the vector x(down arrow) (x(up arrow)) whose components are a decreasing (increasing) rearrangement of the components of x. We characterize the linear maps that preserve (strongly preserve) one of the majorizations <(omega) or <(omega). Copyright (C) 2007.