Linear Maps which Preserve or Strongly Preserve Weak Majorization
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Research Article Linear Maps which Preserve or Strongly Preserve Weak Majorization Ahmad Mohammad Hasani and Mohammad Ali Vali Received 8 July 2007; Accepted 5 November 2007 Dedicated to Professor Mehdi Radjabalipour Recommended by Jewgeni H. Dshalalow
For x, y ∈ Rn , we say x is weakly submajorized (weakly supermajorized) by y, and write x ≺ω y (x ≺ω y), if k1 x[i] ≤ k1 y[i] , k = 1,2,...,n ( k1 x(i) ≥ k1 y(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 . Copyright © 2007 A. M. Hasani and M. A. Vali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The classical majorization and matrix majorization have received considerable attention by many authors. Recently, much interest has focused on the structure of linear preservers and strongly linear preservers of vector and matrix majorizations. Many nice results have been found by Beasley and S. G. Lee [1–4], Ando [5], Dahl [6], Li and Poon [7], and Hasani and Radjabalipour [8–10]. Marshal and Olkin’s text [11] is the standard general reference for majorization. A matrix D with nonnegative entries is called doubly stochastic if the sum of each row of D and also the sum of each row of Dt are 1. Let the following notations be fixed throughout the paper: Mnm (Mm ) for the set of real n × m (m × m) matrices, DS(n) for the set of all n × n doubly stochastic matrices, P(n) for 1 (column) vectors the set of all n × n permutation matrices, Rn for the set of all real n × (note that Rn = Mn1 ), {e1 ,e2 ,...,en } for the standard basis for Rn , e = nj=1 e j , J = eet , the n × n matrix with all entries equal to 1, trx for the trace of the vector x. For x, y ∈ Rn , we say x is weakly submajorized (weakly supermajorized) by y, and we write x ≺ ω y (x ≺ ω y) if
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Journal of Inequalities and Applications k
x[i] ≤
1
k
y[i] ,
k
k = 1,2,...,n
1
x(i) ≥
k
1
y(i) , k = 1,2,...,n ,
(1.1)
1
where x[i] (x(i) ) denotes the ith component of the vector x↓ (x↑ ) whose components are a decreasing (increasing) rearrangement of the components of x. If in addition to x ≺ ω y we also have n1 x j = n1 y j , we say x is majorized by y and write x ≺ y. This definition x ≺ y is equivalent to x = Dy for some D ∈ DS(n) [11]. Given X,Y ∈ Mn,m , we say X is multivariate majorized by Y (written X ≺ Y ) if X = DY for some D ∈ DS(n). When m = 1, the definition of multivariate majorization reduces to the classical concept of majorization on Rn . Let T be a linear map and let R be a relation on Rn . We say T preserves R when R(x, y) implies R(Tx,T y); if in addition R(Tx,T y) implies R(x, y), we say T strongly preserves R. We need the following interesting theorem in ou
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