A characterisation of anti-Löwner functions

Abstract:

According to a celebrated result by Löwner, a real-valued function $f$ is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences $L_f=\left (\frac {f(x_i)-f(x_j)}{x_i-x_j}\right )_{i,j=1}^N$, is positive semidefinite for every integer $N>0$ and any choice of $x_1,x_2,\ldots ,x_N$. In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions $g$ defined on $(0,+\infty )$ for which the matrix of divided sums $K_g=\left (\frac {g(x_i)+g(x_j)}{x_i+x_j}\right )_{i,j=1}^N$, which we call its anti-Löwner matrix, is positive semidefinite for every integer $N>0$ and any choice of $x_1,x_2,\ldots ,x_N\in (0,+\infty )$. Such functions, which we call anti-Löwner functions, have applications in the theory of Lyapunov-type equations.