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Hankel operators on the Bergman space of bounded symmetric domains


Author: Ke He Zhu
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
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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}$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1093426-6
Article copyright: © Copyright 1991 American Mathematical Society

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