Accretive Partial Transpose Matrices and Their Connections to Matrix Means

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 $n\times n$ complex matrices such that $A,B$ are sectorial with sector angle $\alpha$ for some $\alpha\in [0,\pi/2)$, and if $f:(0,\infty)\to(0,\infty)$ is a certain operator monotone function such that $\begin{bmatrix} \cos^2(\alpha) f(A) & X X^* & \cos^2(\alpha) f(B) \end{bmatrix}$ is APT, Then $\begin{bmatrix} f(A)\nabla_t f(B) & X X^* & f(A \nabla_tB ) \end{bmatrix}$ is APT for any $0\leq t\leq 1$, where $\nabla_t$ is the weighted arithmetic mean.