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:
-
Annales Henri Poincaré
- Pub Date:
- April 2024
- DOI:
- 10.1007/s00023-023-01345-7
- arXiv:
- arXiv:2108.05785
- Bibcode:
- 2024AnHP...25.2087Z
- Keywords:
-
- Mathematical Physics;
- Mathematics - Functional Analysis;
- Quantum Physics
- E-Print:
- 16 pages. Revised based on the report of the referee