Jointly convex mappings related to Lieb’s theorem and Minkowski type operator inequalities

• Published in: Analysis and mathematical physics 2021-06, Vol.11 (2), Article 72
• Main Authors: Kian, Mohsen, Seo, Yuki
• Format: Article
• Language: English
• Subjects: Analysis; Concavity; Convexity; Inequalities; Mathematical analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Matrix methods; Operators (mathematics); Tensors; Theorems
• ISSN: 1664-2368; 1664-235X
• DOI: 10.1007/s13324-021-00513-4

Employing the notion of operator log-convexity, we study joint concavity/convexity of multivariable operator functions related to Lieb’s theorem: ( A , B ) ↦ F ( A , B ) = ψ Φ ( f ( A ) ) σ Ψ ( g ( B ) ) , where Φ and Ψ are positive linear mappings and σ is an operator mean. As applications, we prove jointly concavity/convexity of matrix trace functions Tr F ( A , B ) . Moreover, considering positive multi-linear mappings in F ( A ,  B ), our study of the joint concavity/convexity of ( A 1 , … , A k ) ↦ ψ Φ ( f ( A 1 ) , … , f ( A k ) ) provides some generalizations and complement to results of Ando and Lieb concerning the concavity/convexity of mappings involving tensor product. In addition, we present Minkowski type operator inequalities for a unital positive linear mapping, which is an operator version of Minkowski type matrix trace inequalities under a more general setting than Carlen and Lieb, Bekjan, and Hiai.