Accretive Partial Transpose Matrices and Their Connections to Matrix Means (2503.09875v1)

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 $n\times n$ 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.