Some convexity and monotonicity results of trace functionals

• URL: http://arxiv.org/abs/2108.05785v2
• Date: Sat, 8 Jul 2023 22:12:35 GMT
• Title: Some convexity and monotonicity results of trace functionals
• Authors: Haonan Zhang
• Abstract summary: We prove the convexity of trace functionals $$(A,B,C)mapsto textTr|BpACq|s,$$ for parameters $(p,q,s)$ that are best possible. We extend some results in citeHP12quasi,CFL16some and resolve a conjecture in citeRZ14 in the matrix setting.
• Score: 1.90365714903665
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible, where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any $n$-by-$n$ matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in \cite{HP12quasi,CFL16some} and resolve a conjecture in \cite{RZ14} in the matrix setting. Other conjectures in \cite{RZ14} will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems.

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