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| 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.<ref name="SSV"/>
| | #REDIRECT [[operator monotone function]] |
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| ==Representation==
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| A function $f$ is operator monotone if and only if
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| \[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \]
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| for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.
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| ==Examples==
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| The following functions are operator monotone:
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| * $z^s$ for $s \in [0, 1]$,
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| * $\log(1 + z)$,
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| * $\frac{z}{r + z}$ for $r > 0$.
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| ==Properties==
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| If $f, f_1, f_2$ are operator monotone, then the following functions are also operator monotone:
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| * $c_1 f_1(z) + c_2 f_2(z)$ for $c_1, c_2 > 0$,
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| * $((f_1(z))^s + (f_2(z))^s)^{1/s}$ and $(f_1(z^s) + f_2(z^s))^{1/s}$ for $s \in (0, 1]$,
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| * $(f_1(z))^s (f_2(z))^{1-s}$ and $f_1(z^s) f_2(z^{1-s})$ for $s \in [0, 1]$,
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| * $f_1(f_2(z))$,
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| * $z^{1-s} f(z^s)$ and $(f(z^s))^{1/s}$ for $s \in (0, 1]$,
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| * $z / f(z)$, $z f(1/z)$ and $1 / f(1/z)$.
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| ==Relation to Bernstein functions==
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| Operator monotone functions form a subclass of [[Bernstein function]]s. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$:
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| \[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
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| has a [[completely monotone function|completely monotone]] density function. In this case
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| \[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \]
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| This explains the name complete Bernstein functions.
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| ==Holomorphic extension==
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| Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that
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| \begin{align*}
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| \Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\
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| f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\
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| \Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 .
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| \end{align*}
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| Conversely, any function $f$ with above properties is an operator monotone function.
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| Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.
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| ==Operator monotone functions of the Laplacian==
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| 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
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| \[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + z) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \]
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| for some $a, b \ge 0$ and $k(z)$ of the form
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| \begin{align*}
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| 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) \\
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| &= \frac{1}{(2 \pi)^{n/2}} \int_0^\infty \left(\frac{\sqrt{r}}{|z|}\right)^{n/2 - 1} K_{n/2 - 1}(\sqrt{r} |z|) \rho(\mathrm d r) .
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| \end{align*}
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| Here $K_\nu$ is the modified Bessel function of the second kind.
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| For $n = 1$, the above expression simplifies to
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| \[ k(z) = \int_0^\infty \frac{e^{-\sqrt{r} |z|}}{2 \sqrt{r}} \, \rho(\mathrm d r) ; \]
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| that is, $k(z)$ can be an arbitrary [[completely monotone function]] of $|z|$, which satisfies the ususal integrability condition $\int_{-\infty}^\infty \min(1, z^2) k(z) dz < \infty$. In a similar way, for $n = 3$,
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| \[ k(z) = \frac{1}{4 \pi |z|} \int_0^\infty e^{-\sqrt{r} |z|} \rho(\mathrm d r) ; \]
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| hence, $|z| k(z)$ can be an arbitrary [[completely monotone function]] of $|z|$, provided that $\int_{\R^3} \min(1, |z|^2) k(z) dz < \infty$.
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| ==References==
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| {{reflist|refs=
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| <ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
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| }}
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| {{stub}}
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