On Araki-Type Trace Inequalities

• URL: http://arxiv.org/abs/2507.05242v1
• Date: Mon, 07 Jul 2025 17:51:50 GMT
• Title: On Araki-Type Trace Inequalities
• Authors: Po-Chieh Liu, Hao-Chung Cheng,
• Abstract summary: We prove a trace inequality $textTr[ f(A) As Bs ] leq textTr[ g(A) (A1/2 B A1/2 )s ]$.
• Score: 6.675805308519987
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we prove a trace inequality $\text{Tr}[ f(A) A^s B^s ] \leq \text{Tr}[ f(A) (A^{1/2} B A^{1/2} )^s ]$ for any positive and monotone increasing function $f$, $s\in[0,1]$, and positive semi-definite matrices $A$ and $B$. On the other hand, for $s\in[0,1]$ such that the map $x\mapsto x^s g(x)$ is positive and decreasing, then $ \text{Tr}[ g(A) (A^{1/2} B A^{1/2} )^s ] \leq \text{Tr}[ g(A) A^s B^s ]$.

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