Operator monotone function

A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.

Representation

A function $f$ is operator monotone if and only if \[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \] for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.

Relation to Bernstein functions

Operator monotone functions form a subclass of Bernstein functions. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$: \[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \] has a completely monotone density function. In this case \[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \] This explains the name complete Bernstein functions.

Holomorphic extension

Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that \begin{align*} \Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\ f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\ \Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 . \end{align*} Conversely, any function $f$ with above properties is an operator monotone function.

Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.

Operator monotone functions of the Laplacian

Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if \[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \] for some $a, b \ge 0$ and $k(z)$ of the form \begin{align*} k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r) \end{align*}

References

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