Arkiv för Matematik

  • Ark. Mat.
  • Volume 30, Number 1-2 (1992), 227-243.

Density of algebras generated by Toeplitz operators on Bergman spaces

Miroslav Engliš

Full-text: Open access

Abstract

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the C*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces A2(Ω) for a wide class of plane domains Ω⊂C, and in Fock spaces A2(CN), N≧1.

Article information

Source
Ark. Mat. Volume 30, Number 1-2 (1992), 227-243.

Dates
Received: 5 May 1991
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898067

Digital Object Identifier
doi:10.1007/BF02384872

Zentralblatt MATH identifier
0784.46036

Rights
1992 © Institut Mittag-Leffler

Citation

Engliš, Miroslav. Density of algebras generated by Toeplitz operators on Bergman spaces. Ark. Mat. 30 (1992), no. 1-2, 227--243. doi:10.1007/BF02384872. https://projecteuclid.org/euclid.afm/1485898067


Export citation

References

  • Axler, S., Bergman spaces and their operators, in: Conway, J. B. and Morrel, B. B. (editors), Survey of some recent results in operator theory, vol. I, 1–50, Pitman Research Notes in Mathematics 171, Longman Scientific & Technical, 1988.
  • Axler, S., Conway, J. B. and McDonald, G., Toeplitz operators on Bergman spaces, Canad. J. Math. 34 (1982), 466–483.
  • Barría, J. and Halmos, P. R., Asymptotic Toeplitz operators, Transactions Amer. Math. Soc. 273 (1982), 621–630.
  • Berezin, F. A., Covariant and contravariant symbols of operators, Math. USSR Izvestiya 6 (1972), 1117–1151.
  • Berezin, F. A., Quantization, Math. USSR Izvestiya 8 (1974), 1109–1163.
  • Berezin, F. A., Quantization in complex symmetric spaces, Math. USSR Izvestiya 9 (1975), 341–379.
  • Berger, C. A. and Coburn, L. A., Toeplitz operators on the Segal-Bargmann space, Trans. AMer. Math. Soc. 301 (1987), 813–829.
  • Berger, C. A. and Coburn, L. A., Berezin-Toeplitz estimates. Preprint.
  • Berger, C. A., Coburn, L. A. and Zhu, K., Function theory on Cartan domains and the Berezin-Teoplitz symbol calculus, Amer. J. Math. 110 (1988), 921–953.
  • Brown, L. G., Douglas, R. G. and Fillmore, P. A., Unitary equivalence modulo the compact operators and extensions of C*-algebras, in: Lecture Notes in Mathematics vol. 345, 58–128. Springer, 1973.
  • Coburn, L. A., Toeplitz operators, quantum mechanics and mean oscillation in the Bergman metric, Proc. Symp. Pure Math. 51 (1990), part I, 97–104.
  • Davidson, K. R., On operators commuting with Toeplitz operators modulo the compact operators, J. Funct. Anal. 24 (1977), 291–302.
  • Engliš, M., A note on Toeplitz operators on Bergman spaces, Comm. Math. Univ. Carolinae 29 (1988), 217–219.
  • Engliš, M., Some density theorems for Toeplitz operators on Bergman spaces, Czech. Math. Journal 40 (115) (1990), 491–502.
  • Folland, G. B., Harmonic analysis in phase space, Annals of Mathematics Studies, Princeton University Press, 1989.
  • Guillemin, V., Toeplitz operators in n-dimensions, Integral Equations Operator Theory 7 (1984), 145–205.
  • Halmos, P. R., A Hilbert space problem book, Graduate texts in mathematics, Springer, 1967.
  • Johnson, B. E. and Parrot, S. K., On operators commuting with a von Neumann algebra modulo the compact operators, J. Funct. Anal. 11 (1972), 39–61.
  • McDonald, G. and Sundberg, C., Toeplitz operators on the disc. Indiana Univ. Math. J. 28 (1979), 595–611.
  • Stroethoff, K., Compact Hankel operators on the Bergman space. Illinois J. Math. 34 (1990), 159–174.
  • Stroethoff, K. and Zheng, D., Toeplitz and Hankel operators in Bergman spaces. Preprint.
  • Zheng, D., Hankel operators and Toeplitz operators on the Bergman space, J. Funct. Anal. 83 (1989), 98–120.