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On the derivatives of the Berezin transform
Authors:
Miroslav Englis and Genkai Zhang
Journal:
Proc. Amer. Math. Soc. 134 (2006), 2285-2294
MSC (2000):
Primary 47B32; Secondary 32A36, 32M15
Posted:
February 2, 2006
MathSciNet review:
2213701
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Additional Information
Abstract: Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator  on the Segal-Bargmann space, the Berezin transform of  is a function whose partial derivatives of all orders are bounded. Similarly, if  is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined ``invariant derivatives'' of any order of the Berezin transform of  are bounded. Further generalizations are also discussed.
References
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J. Arazy, H. Upmeier: Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains, Function spaces, interpolation theory, and related topics (Lund, 2000) (A. Kufner, M. Cwikel, M. Engliš, L.-E. Persson, and G. Sparr, editors), pp. 151-211, Walter de Gruyter, Berlin, 2002. MR 1943284 (2003k:32031)
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L.A. Coburn: A Lipschitz estimate for Berezin's operator calculus, Proc. Amer. Math. Soc. 133 (2005), 127-131. MR 2085161 (2005e:47060)
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M. Engliš: Berezin-Toeplitz quantization on the Schwartz space of bounded symmetric domains, J. Lie Theory 15 (2005), 27-50.MR 2115226
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R. Gangolli, V.S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups, Springer-Verlag, Berlin-Heidelberg, 1988. MR 0954385 (89m:22015)
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I.C. Gohberg, M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs 18, Amer. Math. Soc., Providence, 1969. MR 0246142 (39:7447)
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E. Hille, R.S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications 31, Amer. Math. Soc., Providence, 1957. MR 0089373 (19:664d)
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K. Nam, D. Zheng, C. Zhong:
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Additional Information
Miroslav Englis
Affiliation:
Mathematics Institute, Academy of Sciences of the Czech Republic, Zitná 25, 11567 Praha 1, Czech Republic
Email:
englis@math.cas.cz
Genkai Zhang
Affiliation:
Chalmers Tekniska Högskola/Göteborgs Universitet, 412 96 Göteborg, Sweden
Email:
genkai@math.chalmers.se
DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08238-4
PII:
S 0002-9939(06)08238-4
Keywords:
Bergman kernel,
Berezin transform,
bounded symmetric domain,
invariant differential operator
Received by editor(s):
December 23, 2004
Received by editor(s) in revised form:
March 1, 2005
Posted:
February 2, 2006
Additional Notes:
The research of the first author was supported by GA AV~CR grant no. A1019304
The research of the second author was supported by the Swedish Science Council~(VR)
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2006 American Mathematical Society
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