The Structure of linear preservers of left matrix majorization on R^T

Abstract

Abstract. For vectors X, Y ∈ R n , Y is said to be left matrix majorized by X (Y ≺ X) if for some row stochastic matrix R, Y = RX. A linear operator T : R p → R n is said to be a linear preserver of ≺ if Y ≺ X on R p implies that T Y ≺ T X on R n .

Convex majorization on discrete ℓp spaces

“…Here ≺ l stands for the notion of "left matrix majorization" on R n , which is introduced by F. Khalooei and M. Radjabalipour in [6,7]. Therefore, in this case, to find the structure of isotonic operators on p (I), it suffices to determine the structure of linear operators T : R n → R n , preserving left matrix majorization.…”

“…Therefore, in this case, to find the structure of isotonic operators on p (I), it suffices to determine the structure of linear operators T : R n → R n , preserving left matrix majorization. But, according to [6,7] a linear map T : R n → R n preserves ≺ l , if and only if one of the following conditions hold.…”

“…In [1], Ando characterized the linear operators which preserve vector majorization. On the basis of row-stochastic matrices, Khalooei, Radjabalipour and Salemi [6,7], introduced the concept of left matrix majorization and determined all linear operator preserving left matrix majorization. It is easy to see that for each x, y ∈ R n , x is left matrix majorized by y, if and only if co(x) ⊆ co(y) ( 1 ) Throughout this work, I is a nonempty set, p ∈ [1, +∞), and p (I) is the Banach space of all functions f : I → R with the norm defined by…”

Characterization of two-sided order preserving of convex majorization on lp(I)

“…Majorization theory plays an important role in various areas and gives a lot of applications in the operator theory and linear algebra. For an account of the majorization theory we refer the reader to [1,2,[4][5][6][7][8][9].…”

GUT-majorization and its linear preservers

“…For instance, many norm inequalities result from majorization relations among eigenvalues and singular values of matrices; see, e.g., [1] and [7]. One of the directions is the study of linear functions that preserve or strongly preserve majorization on a space of matrices; see, e.g., [2]- [6]. In this paper, we pay attention to a new kind of majorization which has been defined by a special type of the triangular matrices and we find its linear preservers.…”