Some convexity and monotonicity results of trace functionals
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 DOI:
 10.48550/arXiv.2108.05785
 arXiv:
 arXiv:2108.05785
 Bibcode:
 2021arXiv210805785Z
 Keywords:

 Mathematical Physics;
 Mathematics  Functional Analysis;
 Quantum Physics
 EPrint:
 16 pages. Revised based on the report of the referee