Superadditivity and derivative of operator functions
\textit{M. S. Moslehian} and \textit{H. Najafi} [Integral Equations Oper. Theory 71, No. 4, 575--582 (2011; Zbl 1272.47026)] showed that, if \(A\) and \(B\) are positive operators, then \(AB+BA\geq 0\) if and only if \(f(A+B)\leq f(A)+f(B)\) for every operator monotone function \(f\) on \([0,\infty)\). In the present work, this result is generalized by showing that, if \(A_i\geq 0\) \((i=1,\dots,n)\), then \(\sum_{i\neq j}A_iA_j\geq 0\) if and only if \(f(A_1+\dots+A_n)\leq f(A_1)+\cdots + f(A_n)\) for every non-negative operator monotone function \(f\) on \([0,\infty)\). Moreover, the authors present some other conditions equivalent to the positivity of \(AB+BA\) using the Fréchet derivatives of operator convex functions.
scientific article; zbMATH DE number 6370320
scientific article