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.
Categories: math-ph math.FA math.MP quant-ph