## A product expansion for Toeplitz operators on the Fock space

HTML articles powered by AMS MathViewer

- by Raffael Hagger PDF
- Proc. Amer. Math. Soc.
**147**(2019), 4823-4833 Request permission

## Abstract:

We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results we require significantly fewer derivatives and get the expansion to arbitrary order. This, in particular, improves a result of Borthwick related to Toeplitz quantization. In addition, we derive an intertwining identity between the Berezin star product and the sharp product.## References

- Patrick Ahern and Željko Čučković,
*A theorem of Brown-Halmos type for Bergman space Toeplitz operators*, J. Funct. Anal.**187**(2001), no. 1, 200–210. MR**1867348**, DOI 10.1006/jfan.2001.3811 - Tatyana Barron, Xiaonan Ma, George Marinescu, and Martin Pinsonnault,
*Semi-classical properties of Berezin-Toeplitz operators with $\scr {C}^k$-symbol*, J. Math. Phys.**55**(2014), no. 4, 042108, 25. MR**3390584**, DOI 10.1063/1.4870869 - Wolfram Bauer,
*Berezin-Toeplitz quantization and composition formulas*, J. Funct. Anal.**256**(2009), no. 10, 3107–3142. MR**2504520**, DOI 10.1016/j.jfa.2008.10.002 - Wolfram Bauer and Lewis A. Coburn,
*Uniformly continuous functions and quantization on the Fock space*, Bol. Soc. Mat. Mex. (3)**22**(2016), no. 2, 669–677. MR**3544159**, DOI 10.1007/s40590-016-0108-8 - W. Bauer, L. A. Coburn, and R. Hagger,
*Toeplitz quantization on Fock space*, J. Funct. Anal.**274**(2018), no. 12, 3531–3551. MR**3787599**, DOI 10.1016/j.jfa.2018.01.001 - Wolfram Bauer, Raffael Hagger, and Nikolai Vasilevski,
*Uniform continuity and quantization on bounded symmetric domains*, J. Lond. Math. Soc. (2)**96**(2017), no. 2, 345–366. MR**3708954**, DOI 10.1112/jlms.12069 - F. A. Berezin,
*Covariant and contravariant symbols of operators*, Izv. Akad. Nauk SSSR Ser. Mat.**36**(1972), 1134–1167 (Russian). MR**0350504** - F. A. Berezin,
*General concept of quantization*, Comm. Math. Phys.**40**(1975), 153–174. MR**411452**, DOI 10.1007/BF01609397 - F. A. Berezin,
*Quantization in complex symmetric spaces*, Izv. Akad. Nauk SSSR Ser. Mat.**39**(1975), no. 2, 363–402, 472 (Russian). MR**0508179** - David Borthwick,
*Microlocal techniques for semiclassical problems in geometric quantization*, Perspectives on quantization (South Hadley, MA, 1996) Contemp. Math., vol. 214, Amer. Math. Soc., Providence, RI, 1998, pp. 23–37. MR**1601209**, DOI 10.1090/conm/214/02902 - David Borthwick, Andrzej Lesniewski, and Harald Upmeier,
*Nonperturbative deformation quantization of Cartan domains*, J. Funct. Anal.**113**(1993), no. 1, 153–176. MR**1214901**, DOI 10.1006/jfan.1993.1050 - Arlen Brown and P. R. Halmos,
*Algebraic properties of Toeplitz operators*, J. Reine Angew. Math.**213**(1963/64), 89–102. MR**160136**, DOI 10.1007/978-1-4613-8208-9_{1}9 - L. A. Coburn,
*Deformation estimates for the Berezin-Toeplitz quantization*, Comm. Math. Phys.**149**(1992), no. 2, 415–424. MR**1186036**, DOI 10.1007/BF02097632 - Miroslav Engliš,
*Weighted Bergman kernels and quantization*, Comm. Math. Phys.**227**(2002), no. 2, 211–241. MR**1903645**, DOI 10.1007/s002200200634 - Miroslav Engliš,
*An excursion into Berezin-Toeplitz quantization and related topics*, Quantization, PDEs, and geometry, Oper. Theory Adv. Appl., vol. 251, Birkhäuser/Springer, Cham, 2016, pp. 69–115. MR**3494849**, DOI 10.1007/978-3-319-22407-7_{2} - Miroslav Engliš and Harald Upmeier,
*Toeplitz quantization and asymptotic expansions: Peter-Weyl decomposition*, Integral Equations Operator Theory**68**(2010), no. 3, 427–449. MR**2735445**, DOI 10.1007/s00020-010-1808-5 - Miroslav Engliš and Harald Upmeier,
*Toeplitz quantization and asymptotic expansions for real bounded symmetric domains*, Math. Z.**268**(2011), no. 3-4, 931–967. MR**2818737**, DOI 10.1007/s00209-010-0702-9 - Sławomir Klimek and Andrzej Lesniewski,
*Quantum Riemann surfaces. I. The unit disc*, Comm. Math. Phys.**146**(1992), no. 1, 103–122. MR**1163670**, DOI 10.1007/BF02099210 - Xiaonan Ma and George Marinescu,
*Berezin-Toeplitz quantization on Kähler manifolds*, J. Reine Angew. Math.**662**(2012), 1–56. MR**2876259**, DOI 10.1515/CRELLE.2011.133 - Marc A. Rieffel,
*Deformation quantization and operator algebras*, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 411–423. MR**1077400**

## Additional Information

**Raffael Hagger**- Affiliation: Institut für Analysis, Leibniz Universität Hanover, 30167 Hannover, Germany
- MR Author ID: 1116916
- Email: raffael.hagger@math.uni-hannover.de
- Received by editor(s): August 30, 2018
- Received by editor(s) in revised form: February 8, 2019
- Published electronically: June 10, 2019
- Communicated by: Stephan Ramon Garcia
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4823-4833 - MSC (2010): Primary 47B35; Secondary 46L65, 30H20
- DOI: https://doi.org/10.1090/proc/14661
- MathSciNet review: 4011516