Linear maps of positive partial transpose matrices and singular value inequalities

Abstract

Linear maps Φ : M n → M k are called m -PPT if [Φ(A i j )] m i, j=1 are positive partial transpose matrices for all positive semi-definite matricesIn 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 → X + a(tr X)I and X → a(tr X)I − X for a 0 . Moreover, singular values inequalities of 2 × 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.

Improvements on some partial trace inequalities for positive semidefinite block matrices

“…In 2014, Lin [27, Proposition 2.5] proved the 2-copositivity of Φ(X) = (trX)I + X and then obtained the following trace inequality (13). More precisely, if…”

“…Furthermore, if A = [A i,j ] m i,j=1 ∈ M m (M n ) is positive semidefinite, it is easy to see that both tr 1 A and tr 2 A are also positive semidefinite; see, e.g., [38, p. 237] or [39,Theorem 2.1]. Over the years, various results involving partial transpose and partial traces have been obtained in the literature, e.g., [3,9,11,13,24]. We introduce the background and recent progress briefly.…”

“…* B; see[28, Theorem 2.1]. Motivated by this observation, Kittaneh and Lin[19] further generalized(13) to the following (14) by an elegant self-improved technique. trAtrC − |trB| 2 ≥ tr(B * B) − tr(AC).…”