Hankel operators on the Bergman space of bounded symmetric domains
HTML articles powered by AMS MathViewer
- by Ke He Zhu PDF
- Trans. Amer. Math. Soc. 324 (1991), 707-730 Request permission
Abstract:
Let $\Omega$ be a bounded symmetric domain in ${\mathbb {C}^n}$ with normalized volume measure $dV$. Let $P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the Bergman space $L_a^2(\Omega )$ of holomorphic functions in ${L^2}(\Omega ,dV)$. Let $\overline P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the closed subspace of antiholomorphic functions in ${L^2}(\Omega ,dV)$. The "little" Hankel operator ${h_f}$ with symbol $f$ is the operator from $L_a^2(\Omega )$ into ${L^2}(\Omega ,dV)$ defined by ${h_f}g = \overline P (fg)$. We characterize the boundedness, compactness, and membership in the Schatten classes of the Hankel operators ${h_f}$ in terms of a certain integral transform of the symbol $f$. These characterizations are further studied in the special cases of the open unit ball and the poly-disc in ${\mathbb {C}^n}$.References
- J. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), no. 6, 989–1053. MR 970119, DOI 10.2307/2374685
- 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
- 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
- R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^{p}$, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 11–66. MR 604369
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
- Svante Janson, Jaak Peetre, and Richard Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamericana 3 (1987), no. 1, 61–138. MR 1008445, DOI 10.4171/RMI/46
- Daniel H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), no. 2, 345–368. MR 899655, DOI 10.1016/0022-1236(87)90072-3
- 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
- M. Stoll, Mean value theorems for harmonic and holomorphic functions on bounded symmetric domains, J. Reine Angew. Math. 290 (1977), 191–198. MR 437812, DOI 10.1515/crll.1977.290.191
- Ke He Zhu, Duality and Hankel operators on the Bergman spaces of bounded symmetric domains, J. Funct. Anal. 81 (1988), no. 2, 260–278. MR 971880, DOI 10.1016/0022-1236(88)90100-0
- Ke He Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (1988), no. 2, 329–357. MR 1004127
- Ke He Zhu, The Bergman spaces, the Bloch space, and Gleason’s problem, Trans. Amer. Math. Soc. 309 (1988), no. 1, 253–268. MR 931533, DOI 10.1090/S0002-9947-1988-0931533-6
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 324 (1991), 707-730
- MSC: Primary 47B35; Secondary 47B10
- DOI: https://doi.org/10.1090/S0002-9947-1991-1093426-6
- MathSciNet review: 1093426