Some singular value inequalities related to linear maps

Abstract

If (A X X* B) ? M2(Mn) is positive semidefinite, Lin [7] conjectured that 2sj(?(X))?sj(?(A)+ (B)), j=1,..., n, and sj(?(X))? sj(?(A)#?(B)), j = 1,..., n, where the linear map ?:X 7 ? 2tr(X)In-X and sj(?) means the j-th largest singular value. In this paper, we reprove that (?(A) ?(X) ?(X*)?(B)) is PPT by using an alternative approach and prove the above singular value inequalities hold for the linear map ?1 : X ?(2n + 1)tr(X)In-X.