A Generalization of Lieb concavity theorem

Abstract

Lieb concavity theorem, successfully solved the Wigner-Yanase-Dyson conjecture, which is a very important theorem, and there are many proofs of it. Generalization of the Lieb concavity theorem has been obtained by Huang, which implies that it is jointly concave for any nonnegative matrix monotone function $ f(x) $ over $ \left(\operatorname{Tr}\left[\wedge^{k}(A^{\frac{qs}{2}}K^{\ast}B^{sp}KA^{\frac{sq}{2}})^{\frac{1}{s}}\right]\right)^{\frac{1}{k}} $. In this manuscript, we obtained $ \left(\operatorname{Tr}\left[\wedge^{k}(f(A^{\frac{qs}{2}})K^{\ast}f(B^{sp})Kf(A^{\frac{sq}{2}}))^{\frac{1}{s}}\right]\right)^{\frac{1}{k}} $ was jointly concave for any nonnegative matrix monotone function $ f(x) $ by using Epstein's theorem, and some more general results were obtained.

Keywords

• lieb concavity
• the wigner-yanase-dyson conjecture
• nonnegative matrix monotone function
• epstein theorem