Accretive Partial Transpose Matrices and Their Connections to Matrix Means
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917954830991360 |
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| author | Aldabbas, Eman Sababheh, Mohammad |
| author_facet | Aldabbas, Eman Sababheh, Mohammad |
| contents | 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 $α$ for some \(α\in [0,π/2)\), and if \(f:(0,\infty)\to(0,\infty)\) is a certain operator monotone function such that \(\begin{bmatrix}
\cos^2(α) f(A) & X
X^* & \cos^2(α) f(B)
\end{bmatrix}\) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Accretive Partial Transpose Matrices and Their Connections to Matrix Means Aldabbas, Eman Sababheh, Mohammad Functional Analysis 47A63, 47A30, 47A64 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 $α$ for some \(α\in [0,π/2)\), and if \(f:(0,\infty)\to(0,\infty)\) is a certain operator monotone function such that \(\begin{bmatrix} \cos^2(α) f(A) & X X^* & \cos^2(α) f(B) \end{bmatrix}\) 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. |
| title | Accretive Partial Transpose Matrices and Their Connections to Matrix Means |
| topic | Functional Analysis 47A63, 47A30, 47A64 |
| url | https://arxiv.org/abs/2503.09875 |