Accretive partial transpose matrices and their connections to matrix means

Abstract Accretive partial transpose (APT) matrices have been recently defined, as a natural extension of positive partial transpose (PPT) matrices. In this paper, we discuss further properties of APT matrices in a way that extends some of those properties known for PPT matrices. Among many results, we show that if A, B, X are n × n complex matrices such that A, B are sectorial with sector angle α for some α ∈ [0, π/2), and if f : (0, ∞) → (0, ∞) is a certain operator monotone function such that cos 2 ( α ) f ( A ) X X * cos 2 ( α ) f ( B ) $\left[\begin{matrix}\hfill {\mathrm{cos}}^{2}\left(\alpha \right)f\left(A\right)\hfill & \hfill X\hfill \\ \hfill {X}^{{\ast}}\hfill & \hfill {\mathrm{cos}}^{2}\left(\alpha \right)f\left(B\right)\hfill \end{matrix}\right]$ is APT, Then f ( A ) ∇ t f ( B ) X X * f ( A ∇ t B ) $\left[\begin{matrix}\hfill f\left(A\right){\nabla }_{t}f\left(B\right)\hfill & \hfill X\hfill \\ \hfill {X}^{{\ast}}\hfill & \hfill f\left(A{\nabla }_{t}B\right)\hfill \end{matrix}\right]$ is APT for any 0 ≤ t ≤ 1, where A∇ t B is the weighted arithmetic mean of A and B.