Weyl calculus for complex and real symmetric domains

Jonathan Arazy; Harald Upmeier

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 3-4, page 165-181
  • ISSN: 1120-6330

Abstract

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We define the Weyl functional calculus for real and complex symmetric domains, and compute the associated Weyl transform in the rank 1 case.

How to cite

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Arazy, Jonathan, and Upmeier, Harald. "Weyl calculus for complex and real symmetric domains." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.3-4 (2002): 165-181. <http://eudml.org/doc/252432>.

@article{Arazy2002,
abstract = {We define the Weyl functional calculus for real and complex symmetric domains, and compute the associated Weyl transform in the rank 1 case.},
author = {Arazy, Jonathan, Upmeier, Harald},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Functional calculi; Symmetric domains; Weyl transform; Weyl functional calculus; real bounded symmetric domain; weighted Bergman space of holomorphic functions},
language = {eng},
month = {12},
number = {3-4},
pages = {165-181},
publisher = {Accademia Nazionale dei Lincei},
title = {Weyl calculus for complex and real symmetric domains},
url = {http://eudml.org/doc/252432},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Arazy, Jonathan
AU - Upmeier, Harald
TI - Weyl calculus for complex and real symmetric domains
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/12//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 3-4
SP - 165
EP - 181
AB - We define the Weyl functional calculus for real and complex symmetric domains, and compute the associated Weyl transform in the rank 1 case.
LA - eng
KW - Functional calculi; Symmetric domains; Weyl transform; Weyl functional calculus; real bounded symmetric domain; weighted Bergman space of holomorphic functions
UR - http://eudml.org/doc/252432
ER -

References

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  1. Arazy, J. - Upmeier, H., Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains. Conference on Function Spaces, Interpolation Theory, and related topics in honour of Jaak Peetre on his 65th birthday. University of Lund, Sweden, August 17-22 2000. Zbl1027.32020
  2. Arazy, J. - Upmeier, H., Covariant symbolic calculi on real symmetric domains. International Workshop on Operator Theory and Applications (Faro, September 12-15 2000). IWOTA-Portugal2000. Zbl1034.46071
  3. van Dijk, G. - Pevzner, M., Berezin kernels and tube domains. J. Funct. Anal., to appear. Zbl0970.43003MR1821696DOI10.1006/jfan.2000.3706
  4. Faraut, J. - Korányi, A., Analysis on Symmetric Cones. Clarendon Press, Oxford1994. Zbl0841.43002MR1446489
  5. Folland, G., Harmonic Analysis in Phase Space. Princeton Univ. Press, 1989. Zbl0682.43001MR983366
  6. Helgason, S., Groups and Geometric Analysis. Academic Press, New York1984. Zbl0543.58001MR754767
  7. Loos, O., Bounded Symmetric Domains and Jordan Pairs. Univ. of California, Irvine1977. 
  8. Magnus, W. - Oberhettinger, F. - Soni, R.P., Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, New York1966. Zbl0143.08502MR232968
  9. Neretin, Y., Matrix analogs of Beta-integral and Plancherel formula for Berezin kernel representations. Preprint. 
  10. Unterberger, A. - Unterberger, J., La série discrète de S ⁢ L ⁢ 2 , R et les opérateurs pseudo-différentiels sur une demi-droite. Ann. Sci. Ec. Norm. Sup., 17, 1984, 83-116. Zbl0549.35119MR744069
  11. Unterberger, A. - Unterberger, J., A quantization of the Cartan domain B ⁢ D I q = 2 and operators on the light cone. J. Funct. Anal., 72, 1987, 279-319. Zbl0632.58033MR886815DOI10.1016/0022-1236(87)90090-5
  12. Unterberger, A. - Upmeier, H., The Berezin transform and invariant differential operators. Comm. Math. Phys., 164, 1994, 563-597. Zbl0843.32019MR1291245
  13. Upmeier, H., Symmetric Banach Manifolds and Jordan C ∗ -Algebras. North Holland1985. Zbl0561.46032MR776786
  14. Upmeier, H., Weyl quantization of symmetric spaces: hyperbolic matrix domains. J. Funct. Anal., 96, 1991, 297-330. Zbl0736.47014MR1101260DOI10.1016/0022-1236(91)90064-C
  15. Zhang, G., Berezin transform on real bounded symmetric domains. Preprint. Zbl0965.22015MR1837258DOI10.1090/S0002-9947-01-02832-X

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