Berezin transform and Weyl-type unitary operators on the Bergman space

Author:
L. A. Coburn

Journal:
Proc. Amer. Math. Soc. **140** (2012), 3445-3451

MSC (2010):
Primary 47B32; Secondary 32A36

Published electronically:
February 15, 2012

MathSciNet review:
2929013

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For the open complex unit disc with normalized area measure, we consider the Bergman space of square-integrable holomorphic functions on . Induced by the group of biholomorphic automorphisms of , there is a standard family of Weyl-type unitary operators on . For all bounded operators on , the Berezin transform is a smooth, bounded function on . The range of the mapping Ber: is invariant under . The ``mixing properties'' of the elements of are visible in the Berezin transforms of the induced unitary operators. Computations involving these operators show that there is no real number with for all bounded operators and are used to check other possible properties of . Extensions to other domains are discussed.

**1.**V. Bargmann,*On a Hilbert space of analytic functions and an associated integral transform*, Comm. Pure Appl. Math.**14**(1961), 187–214. MR**0157250****2.**D. Békollé, C. A. Berger, L. A. Coburn, and K. H. Zhu,*BMO in the Bergman metric on bounded symmetric domains*, J. Funct. Anal.**93**(1990), no. 2, 310–350. MR**1073289**, 10.1016/0022-1236(90)90131-4**3.**C. A. Berger, L. A. Coburn, and K. H. Zhu,*Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus*, Amer. J. Math.**110**(1988), no. 5, 921–953. MR**961500**, 10.2307/2374698**4.**F. A. Berezin,*Covariant and contravariant symbols of operators*, Izv. Akad. Nauk SSSR Ser. Mat.**36**(1972), 1134–1167 (Russian). MR**0350504****5.**C. A. Berger and J. G. Stampfli,*Mapping theorems for the numerical range*, Amer. J. Math.**89**(1967), 1047–1055. MR**0222694****6.**L. A. Coburn,*A Lipschitz estimate for Berezin’s operator calculus*, Proc. Amer. Math. Soc.**133**(2005), no. 1, 127–131 (electronic). MR**2085161**, 10.1090/S0002-9939-04-07476-3**7.**Jacques Faraut, Soji Kaneyuki, Adam Korányi, Qi-keng Lu, and Guy Roos,*Analysis and geometry on complex homogeneous domains*, Progress in Mathematics, vol. 185, Birkhäuser Boston, Inc., Boston, MA, 2000. MR**1727259****8.**Gerald B. Folland,*Real analysis*, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR**1681462****9.**Sigurdur Helgason,*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561****10.**Max Koecher,*An elementary approach to bounded symmetric domains*, Rice University, Houston, Tex., 1969. MR**0261032**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
47B32,
32A36

Retrieve articles in all journals with MSC (2010): 47B32, 32A36

Additional Information

**L. A. Coburn**

Affiliation:
Department of Mathematics, The State University of New York at Buffalo, Buffalo, New York 14260

Email:
lcoburn@buffalo.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11440-6

Received by editor(s):
April 5, 2011

Published electronically:
February 15, 2012

Communicated by:
Richard Rochberg

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.