Holomorphic retractions and boundary Berezin transforms

Jonathan Arazy[1]; Miroslav Engliš[2]; Wilhelm Kaup[3]

  • [1] University of Haifa Department of Mathematics 31905 Haifa (Israel)
  • [2] Silesian University in Opava Mathematics Institute Na Rybníčku 1 74601 Opava (Czech Republic) and Mathematics Institute Žitná 25 11567 Praha 1 (Czech Republic)
  • [3] Universität Tübingen Mathematisches Institut Auf der Morgenstelle 10 72076 Tübingen (Germany)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 641-657
  • ISSN: 0373-0956

Abstract

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In an earlier paper, the first two authors have shown that the convolution of a function f continuous on the closure of a Cartan domain and a K -invariant finite measure μ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face F depends only on the restriction of f to F and is equal to the convolution, in  F , of the latter restriction with some measure μ F on F uniquely determined by  μ . In this article, we give an explicit formula for μ F in terms of  F , showing in particular that for measures μ corresponding to the Berezin transforms the measures μ F again correspond to Berezin transforms, but with a shift in the value of the Wallach parameter. Finally, we also obtain a nice and simple description of the holomorphic retraction on these domains which arises as the boundary limit of geodesic symmetries.

How to cite

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Arazy, Jonathan, Engliš, Miroslav, and Kaup, Wilhelm. "Holomorphic retractions and boundary Berezin transforms." Annales de l’institut Fourier 59.2 (2009): 641-657. <http://eudml.org/doc/10408>.

@article{Arazy2009,
abstract = {In an earlier paper, the first two authors have shown that the convolution of a function $f$ continuous on the closure of a Cartan domain and a $K$-invariant finite measure $\mu $ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face $F$ depends only on the restriction of $f$ to $F$ and is equal to the convolution, in $F$, of the latter restriction with some measure $\mu _F$ on $F$ uniquely determined by $\mu $. In this article, we give an explicit formula for $\mu _F$ in terms of $F$, showing in particular that for measures $\mu $ corresponding to the Berezin transforms the measures $\mu _F$ again correspond to Berezin transforms, but with a shift in the value of the Wallach parameter. Finally, we also obtain a nice and simple description of the holomorphic retraction on these domains which arises as the boundary limit of geodesic symmetries.},
affiliation = {University of Haifa Department of Mathematics 31905 Haifa (Israel); Silesian University in Opava Mathematics Institute Na Rybníčku 1 74601 Opava (Czech Republic) and Mathematics Institute Žitná 25 11567 Praha 1 (Czech Republic); Universität Tübingen Mathematisches Institut Auf der Morgenstelle 10 72076 Tübingen (Germany)},
author = {Arazy, Jonathan, Engliš, Miroslav, Kaup, Wilhelm},
journal = {Annales de l’institut Fourier},
keywords = {Berezin transform; Cartan domain; convolution operator},
language = {eng},
number = {2},
pages = {641-657},
publisher = {Association des Annales de l’institut Fourier},
title = {Holomorphic retractions and boundary Berezin transforms},
url = {http://eudml.org/doc/10408},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Arazy, Jonathan
AU - Engliš, Miroslav
AU - Kaup, Wilhelm
TI - Holomorphic retractions and boundary Berezin transforms
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 641
EP - 657
AB - In an earlier paper, the first two authors have shown that the convolution of a function $f$ continuous on the closure of a Cartan domain and a $K$-invariant finite measure $\mu $ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face $F$ depends only on the restriction of $f$ to $F$ and is equal to the convolution, in $F$, of the latter restriction with some measure $\mu _F$ on $F$ uniquely determined by $\mu $. In this article, we give an explicit formula for $\mu _F$ in terms of $F$, showing in particular that for measures $\mu $ corresponding to the Berezin transforms the measures $\mu _F$ again correspond to Berezin transforms, but with a shift in the value of the Wallach parameter. Finally, we also obtain a nice and simple description of the holomorphic retraction on these domains which arises as the boundary limit of geodesic symmetries.
LA - eng
KW - Berezin transform; Cartan domain; convolution operator
UR - http://eudml.org/doc/10408
ER -

References

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  9. L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, (1963), American Mathematical Society, Providence, R.I. Zbl0507.32025MR171936
  10. W. Kaup, J. Sauter, Boundary structure of bounded symmetric domains, Manuscripta Math. 101 (2000), 351-360 Zbl0981.32012MR1751038
  11. Ottmar Loos, Bounded symmetric domains and Jordan pairs, (1977) Zbl0228.32012
  12. Jaak Peetre, The Berezin transform and Ha-plitz operators, J. Operator Theory 24 (1990), 165-186 Zbl0793.47026MR1086552
  13. Karl Stein, Maximale holomorphe und meromorphe Abbildungen. II, Amer. J. Math. 86 (1964), 823-868 Zbl0144.33901MR171030
  14. A. Unterberger, H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563-597 Zbl0843.32019MR1291245
  15. Harald Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics, 67 (1987), Published for the Conference Board of the Mathematical Sciences, Washington, DC Zbl0608.17013MR874756
  16. Genkai Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769-3787 (electronic) Zbl0965.22015MR1837258

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