Matrix inequalities from a two variables functional

Abstract

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[Formula: see text] is a positive semidefinite matrix and[Formula: see text] is a normal matrix,[Formula: see text] and[Formula: see text] is a subunital positive linear map, then[Formula: see text] is weakly log-majorized by[Formula: see text]. This far extension of Araki’s theorem (when [Formula: see text] is the identity and [Formula: see text] 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.