Superadditivity and derivative of operator functions

• Published in: Linear algebra and its applications Vol. 465; pp. 401 - 411
• Main Authors: Uchiyama, Mitsuru, Uchiyama, Atsushi, Giga, Mariko
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-01-2015
• Subjects: Fréchet derivative; Matrix order; Operator concave function; Operator convex function; Operator monotone function; Operator concave function; Matrix order; Fréchet derivative; 47A56; Operator monotone function; Operator convex function; 47A63
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2014.09.006

We will show that if ∑i≠jAiAj≥0 for bounded operators Ai≥0 (i=1,2,⋯,n), then g(∑iAi)≥∑ig(Ai) for every operator convex function g(t) on [0,∞) with g(0)≤0; in particular, (∑iAi)log⁡(∑iAi)≥∑iAilog⁡Ai if each Ai is invertible. Let A,B≥0 and A be invertible. Then we will observe that the Fréchet derivative Dg(sA)(B) is increasing on 0<s<∞ for every operator convex function g(t) on (0,∞) if and only if AB+BA≥0.