# Iterates and the boundary behavior of the Berezin transform

Jonathan Arazy^{[1]}; Miroslav Engliš^{[2]}

- [1] University of Haifa, Department of Mathematics, Haifa 31905 (Israël)
- [2] Academy of Sciences, Mathematics Institute, Ztiná 25, 11567 Prague 1 (République Thèque)

Annales de l’institut Fourier (2001)

- Volume: 51, Issue: 4, page 1101-1133
- ISSN: 0373-0956

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topArazy, Jonathan, and Engliš, Miroslav. "Iterates and the boundary behavior of the Berezin transform." Annales de l’institut Fourier 51.4 (2001): 1101-1133. <http://eudml.org/doc/115936>.

@article{Arazy2001,

abstract = {Let $\mu $ be a measure on a domain $\Omega $ in $\{\mathbb \{C\}\}^n$ such that the Bergman space
of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and
$K(x,x)>0$$\forall x\in \Omega $. The Berezin transform associated to $\mu $ is the
integral operator\[ Bf(y) = K(y,y)^\{-1\} \int \_\Omega f(x)\vert K(x,y)\vert ^2 \,d\mu (x).
\]The number $Bf(y)$ can be interpreted as a certain mean value of $f$ around $y$, and
functions satisfying $Bf=f$ as functions having a certain mean-value property. In this
paper we investigate the boundary behavior of $Bf$, the existence of functions $f$ satisfying $Bf=f$ and having prescribed boundary values, and the convergence of the
iterates $B^k f$, $k\rightarrow \infty $. The best results are obtained for smoothly bounded
strictly pseudoconvex domains $\Omega $ with any measure $\mu $ as above, and for bounded
symmetric domains $\Omega $ and $\mu $ one of the standard rotation-invariant measures on
them. We also carry out similar investigation for convolution operators $B_\mu f=f*\mu $ on a bounded symmetric domain $\Omega =G/K$ with a $K$-invariant absolutely continuous
probability measure $\mu $, and study the behavior of the geodesic symmetries $\phi _a$ of
$\Omega $ as $a$ tends to the boundary.},

affiliation = {University of Haifa, Department of Mathematics, Haifa 31905 (Israël); Academy of Sciences, Mathematics Institute, Ztiná 25, 11567 Prague 1 (République Thèque)},

author = {Arazy, Jonathan, Engliš, Miroslav},

journal = {Annales de l’institut Fourier},

keywords = {Berezin transform; geodesic symmetry; Cartan domain; stochastic operator; -Poisson extension; disk algebra; limiting behaviour of the iterates; Bergman space; fixes holomorphic functions; convolution operators},

language = {eng},

number = {4},

pages = {1101-1133},

publisher = {Association des Annales de l'Institut Fourier},

title = {Iterates and the boundary behavior of the Berezin transform},

url = {http://eudml.org/doc/115936},

volume = {51},

year = {2001},

}

TY - JOUR

AU - Arazy, Jonathan

AU - Engliš, Miroslav

TI - Iterates and the boundary behavior of the Berezin transform

JO - Annales de l’institut Fourier

PY - 2001

PB - Association des Annales de l'Institut Fourier

VL - 51

IS - 4

SP - 1101

EP - 1133

AB - Let $\mu $ be a measure on a domain $\Omega $ in ${\mathbb {C}}^n$ such that the Bergman space
of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and
$K(x,x)>0$$\forall x\in \Omega $. The Berezin transform associated to $\mu $ is the
integral operator\[ Bf(y) = K(y,y)^{-1} \int _\Omega f(x)\vert K(x,y)\vert ^2 \,d\mu (x).
\]The number $Bf(y)$ can be interpreted as a certain mean value of $f$ around $y$, and
functions satisfying $Bf=f$ as functions having a certain mean-value property. In this
paper we investigate the boundary behavior of $Bf$, the existence of functions $f$ satisfying $Bf=f$ and having prescribed boundary values, and the convergence of the
iterates $B^k f$, $k\rightarrow \infty $. The best results are obtained for smoothly bounded
strictly pseudoconvex domains $\Omega $ with any measure $\mu $ as above, and for bounded
symmetric domains $\Omega $ and $\mu $ one of the standard rotation-invariant measures on
them. We also carry out similar investigation for convolution operators $B_\mu f=f*\mu $ on a bounded symmetric domain $\Omega =G/K$ with a $K$-invariant absolutely continuous
probability measure $\mu $, and study the behavior of the geodesic symmetries $\phi _a$ of
$\Omega $ as $a$ tends to the boundary.

LA - eng

KW - Berezin transform; geodesic symmetry; Cartan domain; stochastic operator; -Poisson extension; disk algebra; limiting behaviour of the iterates; Bergman space; fixes holomorphic functions; convolution operators

UR - http://eudml.org/doc/115936

ER -

## References

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- S.G. Krantz, Function theory of several complex variables, (1992), Wadsworth & Brooks/Cole, Pacific Grove Zbl0776.32001MR1162310
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