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, theC *-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 spacesA 2(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA 2(C N),N≧1.
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References
Axler, S., Bergman spaces and their operators, in:Conway, J. B. andMorrel, B. B. (editors),Survey of some recent results in operator theory, vol.I, 1–50, Pitman Research Notes in Mathematics171, Longman Scientific & Technical, 1988.
Axler, S., Conway, J. B. andMcDonald, G., Toeplitz operators on Bergman spaces,Canad. J. Math. 34 (1982), 466–483.
Barría, J. andHalmos, 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. andCoburn, L. A., Toeplitz operators on the Segal-Bargmann space,Trans. AMer. Math. Soc. 301 (1987), 813–829.
Berger, C. A. andCoburn, L. A., Berezin-Toeplitz estimates. Preprint.
Berger, C. A., Coburn, L. A. andZhu, 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. andFillmore, 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), partI, 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 inn-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. andParrot, S. K., On operators commuting with a von Neumann algebra modulo the compact operators,J. Funct. Anal. 11 (1972), 39–61.
McDonald, G. andSundberg, 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. andZheng, 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.
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Engliš, M. Density of algebras generated by Toeplitz operators on Bergman spaces. Ark. Mat. 30, 227–243 (1992). https://doi.org/10.1007/BF02384872
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DOI: https://doi.org/10.1007/BF02384872