Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the Berezin-Toeplitz calculus


Author: L. A. Coburn
Journal: Proc. Amer. Math. Soc. 129 (2001), 3331-3338
MSC (2000): Primary 47B35; Secondary 47B32
DOI: https://doi.org/10.1090/S0002-9939-01-05917-2
Published electronically: March 29, 2001
MathSciNet review: 1845010
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex $n$-space, $\mathbf{C}^n$. For several interesting algebras of functions on $\mathbf{C}^n$, we have $T_\varphi T_\psi =T_{\varphi \diamond \psi }$for all $\varphi ,\psi $ in the algebra, where $T_\varphi $ is the Berezin-Toeplitz operator associated with $\varphi $ and $\varphi \diamond \psi $ is a ``twisted'' associative product on the algebra of functions. On the other hand, there is a $C^\infty $ function $\varphi $ for which $T_\varphi $ is bounded but $T_\varphi T_\varphi \neq T_\psi $ for any $\psi$.


References [Enhancements On Off] (What's this?)

  • [B] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure and Appl. Math. 14 (1961), 187-214. MR 28:486
  • [Be] F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134-1167. MR 50:2996
  • [BC1] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. AMS 301 (1987), 813-829. MR 88c:47044
  • [BC2] -, Heat flow and Berezin-Toeplitz estimates, Amer. J. Math. 116 (1994), 563-590. MR 95g:47038
  • [C] L. A. Coburn, The measure algebra of the Heisenberg group, J. Funct. Analysis 161 (1999), 509-525. MR 2000b:46128
  • [F] G. B. Folland, Harmonic analysis in phase space, Annals of Math. Studies, Princeton Univ. Press, Princeton, N.J., 1989.MR 92k:22017
  • [G] M. Gerstenhaber, On the deformation of rings and algebras, III, Annals. of Math. (2) 88 (1968), 1-34. MR 39:1521
  • [Gu] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations and Operator Theory 7 (1984), 145-205. MR 86i:58130
  • [KL] S. Klimek and A. Lesniewski, Quantum Riemann surfaces, I. The unit disc, Comm. Math. Phys. 146 (1992) 103-122. MR 93g:58009

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B35, 47B32

Retrieve articles in all journals with MSC (2000): 47B35, 47B32


Additional Information

L. A. Coburn
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: lcoburn@acsu.buffalo.edu

DOI: https://doi.org/10.1090/S0002-9939-01-05917-2
Received by editor(s): December 21, 1999
Received by editor(s) in revised form: March 21, 2000
Published electronically: March 29, 2001
Additional Notes: The author’s research was supported by a grant of the NSF and a visiting membership in the Erwin Schrödinger Institute.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society