Template:High-risk and Operator monotone function: Difference between pages

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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"/>
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{{Documentation}}
==Representation==
<!-- Add categories and interwikis to the /doc subpage, not here! -->
A function $f$ is operator monotone if and only if
</noinclude>
\[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \]
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 function]]s. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$:
\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
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==
{{reflist|refs=
<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>
}}
 
{{stub}}

Revision as of 03:34, 19 July 2012

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.[1]

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

  1. Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311 

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