LINEAR MAPS OF POSITIVE PARTIAL TRANSPOSE MATRICES AND SINGULAR VALUE INEQUALITIES

Linear maps Phi : M-n -> M-k are called m-PPT if [Phi(A(ij))](i,j=1)(m) are positive partial transpose matrices for all positive semi-definite matrices [A(ij)](i,j=1)(m) is an element of M-m(M-n). In this paper, connections between m-PPT maps, m -positive maps and m-copositive maps are given. In consequence, characterizations of completely PPT maps are obtained. The results are applied to study two linear maps X bar right arrow X + a(trX)I and X bar right arrow a(trX)I - X for a >= 0. Moreover, singular values inequalities of 2 x 2 positive block matrices under these two linear maps are given. In particular, we prove an open singular values inequality formulated by Lin [Linear Algebra Appl, 520 (2017)] for n <= 3.