Around operator monotone functions
There has been considerable interest in obtaining sub-additive (super-additive) inequalities for matrix functions analogous to the sub-additive (super-additive) inequality for concave (convex) functions of the reals. If the function \(f:[0,\infty )\rightarrow [0,\infty )\) is operator concave, then the sub-additive inequality \(f(A+B)\leq f(A)+f(B)\) does not hold in general for positive semidefinite \(n\times n\) matrices \(A,B\). In this paper, the authors prove that this sub-additive inequality holds if and only if the symmetrized product \(AB+BA\) is positive semidefinite. The authors also give a necessary and sufficient condition that the composition \(f\circ g\) of an operator convex function \(f\) on \([0,\infty )\) and a non-negative operator monotone function \(g\) on an interval \((a,b)\) is operator monotone. Some applications are also given.
scientific article; zbMATH DE number 6048412