Löwner's theorem

Remarks

So an operator monotone function is real analytic, and it extends to a holomorphic map from $\mathbb{H}$ into the closed upper half-plane, a so-called Pick function. Example: for $0 < p < 1$, the function $t^p$ on $(0, \infty)$ extends to $z^p$ with the principal branch. It maps $\mathbb{H}$ into the sector of arguments between $0$ and $p\pi$, which lies in $\mathbb{H}$, so $t^p$ is operator monotone. In the same way $\log t$ is operator monotone on $(0, \infty)$, since $\log z$ maps $\mathbb{H}$ into the strip $0 < \operatorname{Im} w < \pi$. In contrast, $z^2$ maps $-1 + i \in \mathbb{H}$ to $-2i$, so $t^2$ is not operator monotone on $(0, \infty)$. One route to the proof goes through divided differences. A function $f$ of class $C^1$ is operator monotone if and only if the following holds for all points $t_1, \dots, t_n$ of $(a, b)$. The matrix with entries $(f(t_i) - f(t_j)) / (t_i - t_j)$ off the diagonal and $f'(t_i)$ on the diagonal is positive semidefinite.

Going deeper

For a student

The theorem gives integral representations of operator monotone functions, through the Nevanlinna representation of Pick functions. It explains at once why $t^p$ is operator monotone on $(0, \infty)$ exactly for $0 \le p \le 1$. Operator monotone functions appear in matrix inequalities, in the theory of operator means and in quantum information.