## On the Berezin-Toeplitz calculus

HTML articles powered by AMS MathViewer

- by L. A. Coburn PDF
- Proc. Amer. Math. Soc.
**129**(2001), 3331-3338 Request permission

## 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 â€śtwistedâ€ť 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

- V. Bargmann,
*On a Hilbert space of analytic functions and an associated integral transform*, Comm. Pure Appl. Math.**14**(1961), 187â€“214. MR**157250**, DOI 10.1002/cpa.3160140303 - F. A. Berezin,
*Covariant and contravariant symbols of operators*, Izv. Akad. Nauk SSSR Ser. Mat.**36**(1972), 1134â€“1167 (Russian). MR**0350504** - C. A. Berger and L. A. Coburn,
*Toeplitz operators on the Segal-Bargmann space*, Trans. Amer. Math. Soc.**301**(1987), no.Â 2, 813â€“829. MR**882716**, DOI 10.1090/S0002-9947-1987-0882716-4 - C. A. Berger and L. A. Coburn,
*Heat flow and Berezin-Toeplitz estimates*, Amer. J. Math.**116**(1994), no.Â 3, 563â€“590. MR**1277446**, DOI 10.2307/2374991 - L. A. Coburn,
*The measure algebra of the Heisenberg group*, J. Funct. Anal.**161**(1999), no.Â 2, 509â€“525. MR**1674651**, DOI 10.1006/jfan.1998.3354 - Gerald B. Folland,
*Harmonic analysis in phase space*, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR**983366**, DOI 10.1515/9781400882427 - Murray Gerstenhaber,
*On the deformation of rings and algebras. III*, Ann. of Math. (2)**88**(1968), 1â€“34. MR**240167**, DOI 10.2307/1970553 - Victor Guillemin,
*Toeplitz operators in $n$ dimensions*, Integral Equations Operator Theory**7**(1984), no.Â 2, 145â€“205. MR**750217**, DOI 10.1007/BF01200373 - SĹ‚awomir Klimek and Andrzej Lesniewski,
*Quantum Riemann surfaces. I. The unit disc*, Comm. Math. Phys.**146**(1992), no.Â 1, 103â€“122. MR**1163670**

## 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
- 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 authorâ€™s research was supported by a grant of the NSF and a visiting membership in the Erwin SchrĂ¶dinger Institute.
- Communicated by: Joseph A. Ball
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1845010