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 $\alpha \in [0,\pi \mathrm{/}2)$, and if $f:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is a certain operator monotone function such that $\left[\begin{array}{ccc}{\cos }^2(\alpha )f(A)& X{X}^{\ast }& {\cos }^2(\alpha )f(B)\end{array}\right]$ is APT, Then $\left[\begin{array}{ccc}f(A){\mathrm{\nabla }}_{t}f(B)& X{X}^{\ast }& f(A{\mathrm{\nabla }}_{t}B)\end{array}\right]$ is APT for any $0\le t\le 1$, where ${\mathrm{\nabla }}_t$ is the weighted arithmetic mean.