Operator monotone function - Revision history

imported>Mateusz: /* Operator monotone functions of the Laplacian */ explicit description in dimensions 1 and 3

2012-07-23T12:22:22Z

Operator monotone functions of the Laplacian: explicit description in dimensions 1 and 3

← Older revision		Revision as of 07:22, 23 July 2012
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	for some $a, b \ge 0$ and $k(z)$ of the form		for some $a, b \ge 0$ and $k(z)$ of the form
	\begin{align*}		\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)		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) \\
			&= \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) .
	\end{align*}		\end{align*}
			Here $K_\nu$ is the modified Bessel function of the second kind.

			For $n = 1$, the above expression simplifies to
			\[ k(z) = \int_0^\infty \frac{e^{-\sqrt{r} \|z\|}}{2 \sqrt{r}} \, \rho(\mathrm d r) ; \]
			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 $n = 3$,
			\[ k(z) = \frac{1}{4 \pi \|z\|} \int_0^\infty e^{-\sqrt{r} \|z\|} \rho(\mathrm d r) ; \]
			hence, $\|z\| k(z)$ can be an arbitrary [[completely monotone function]] of $\|z\|$, provided that $\int_{\R^3} \min(1, \|z\|^2) k(z) dz < \infty$.

	==References==		==References==

imported>Mateusz: added "Examples" and "Properties"

2012-07-20T12:17:52Z

added "Examples" and "Properties"

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 \[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \]
 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==

imported>Mateusz: typo

2012-07-19T10:28:57Z

typo

← Older revision		Revision as of 05:28, 19 July 2012
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	==Operator monotone functions of the Laplacian==		==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		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
	\[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{\|z\| < 1}) k(z) \mathrm d z \]		\[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + z) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{\|z\| < 1}) k(z) \mathrm d z \]
	for some $a, b \ge 0$ and $k(z)$ of the form		for some $a, b \ge 0$ and $k(z)$ of the form
	\begin{align*}		\begin{align*}

imported>Mateusz: link

2012-07-19T10:14:52Z

link

← Older revision		Revision as of 05:14, 19 July 2012
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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 $f$:		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 $f$:
	\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]		\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
	has a completely monotone density function. In this case		has a [[completely monotone function\|completely monotone]] density function. In this case
	\[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \]		\[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \]
	This explains the name complete Bernstein functions.		This explains the name complete Bernstein functions.

imported>Mateusz: Created page with "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..."

2012-07-19T08:34:55Z

Created page with "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..."

New page

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"/>

==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 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>
}}

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