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arXiv · 2108.05785

Some convexity and monotonicity results of trace functionals

Haonan Zhang
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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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BibTeXRIS

Haonan Zhang. 2021-08-12. Some convexity and monotonicity results of trace functionals. https://doi.org/10.1007/s00023-023-01345-7

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