Toeplitz and Hankel operators on Bergman spaces
HTML articles powered by AMS MathViewer
- by Karel Stroethoff and De Chao Zheng
- Trans. Amer. Math. Soc. 329 (1992), 773-794
- DOI: https://doi.org/10.1090/S0002-9947-1992-1112549-7
- PDF | Request permission
Abstract:
In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in ${\mathbb {C}^n}$ whose symbols are bounded measurable functions. We give necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Toeplitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of ${H^\infty }(\Omega )$; for these symbols we describe when the Toeplitz or Hankel operators are compact.References
- Sheldon Axler, Hankel operators on Bergman spaces, Linear and Complex Analysis Problem Book, edited by V. P. Havin, S. V. Hrušł=c
- Sheldon Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), no. 2, 315–332. MR 850538, DOI 10.1215/S0012-7094-86-05320-2 —, Informal notes on $COP$ and $AOP$, unpublished manuscript.
- Sheldon Axler, John B. Conway, and Gerard McDonald, Toeplitz operators on Bergman spaces, Canadian J. Math. 34 (1982), no. 2, 466–483. MR 658979, DOI 10.4153/CJM-1982-031-1
- Sheldon Axler and Pamela Gorkin, Algebras on the disk and doubly commuting multiplication operators, Trans. Amer. Math. Soc. 309 (1988), no. 2, 711–723. MR 961609, DOI 10.1090/S0002-9947-1988-0961609-9
- D. Békollé, C. A. Berger, L. A. Coburn, and K. H. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), no. 2, 310–350. MR 1073289, DOI 10.1016/0022-1236(90)90131-4
- C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 813–829. MR 882716, DOI 10.1090/S0002-9947-1987-0882716-4
- C. A. Berger, L. A. Coburn, and K. H. Zhu, Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus, Amer. J. Math. 110 (1988), no. 5, 921–953. MR 961500, DOI 10.2307/2374698
- John B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR 634507
- J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), no. 1, 64–89. MR 1033914, DOI 10.1016/0022-1236(90)90119-6
- Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 96, Springer-Verlag, Berlin-New York, 1978. MR 517709
- Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Gerard McDonald, Fredholm properties of a class of Toeplitz operators on the ball, Indiana Univ. Math. J. 26 (1977), no. 3, 567–576. MR 482351, DOI 10.1512/iumj.1977.26.26044
- G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), no. 4, 595–611. MR 542947, DOI 10.1512/iumj.1979.28.28042
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
- Karel Stroethoff, Compact Hankel operators on the Bergman space, Illinois J. Math. 34 (1990), no. 1, 159–174. MR 1031892
- Karel Stroethoff, Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in $\textbf {C}^n$, J. Operator Theory 23 (1990), no. 1, 153–170. MR 1054822
- Karel Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), no. 1, 3–16. MR 1137884, DOI 10.1307/mmj/1029004449 Dechao Zheng, Hankel operators and Toeplitz operators on the Bergman space, J. Funct. Anal. 83 (1989), 98-120.
- De Chao Zheng, Toeplitz operators and Hankel operators, Integral Equations Operator Theory 12 (1989), no. 2, 280–299. MR 986598, DOI 10.1007/BF01195117 —, Semi-commutators of Toeplitz operators on the Bergman space, preprint.
- Ke He Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302 (1987), no. 2, 617–646. MR 891638, DOI 10.1090/S0002-9947-1987-0891638-4
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 329 (1992), 773-794
- MSC: Primary 47B35; Secondary 32A35, 46E20
- DOI: https://doi.org/10.1090/S0002-9947-1992-1112549-7
- MathSciNet review: 1112549