Matrix inequalities from a two variables functional
Résumé
We introduce a two variables norm functional and establish its joint log-convexity. This entails and improves many remarkable matrix inequalities, most of them related to the log-majorization theorem of Araki. In particular: if A is a positive semidefinite matrix and N is a normal matrix, p ≥ 1 and Φ is a sub-unital positive linear map, then |AΦ(N)A| p is weakly log-majorized by A p Φ(|N | p)A p. This far extension of Araki's theorem (when Φ is the identity and N is positive) complements some recent results of Hiai and contains several special interesting cases such as a triangle inequality for normal operators and some extensions of the Golden-Thompson trace inequality. Some applications to Schur products are also obtained.
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