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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

VMO, ESV, and Toeplitz operators on the Bergman space


Author: Ke He Zhu
Journal: Trans. Amer. Math. Soc. 302 (1987), 617-646
MSC: Primary 47B35; Secondary 30H05, 46L99
DOI: https://doi.org/10.1090/S0002-9947-1987-0891638-4
MathSciNet review: 891638
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Abstract: This paper studies the largest $ {C^*}$-subalgebra $ Q$ of $ {L^\infty }({\mathbf{D}})$ such that the Toeplitz operators $ {T_f}$ on the Bergman space $ L_a^2({\mathbf{D}})$ with symbols $ f$ in $ Q$ have a symbol calculus modulo the compact operators. $ Q$ is characterized by a condition of vanishing mean oscillation near the boundary. I also give several other necessary and sufficient conditions for a bounded function to be in $ Q$. After decomposing $ Q$ in a "nice" way, I study the Fredholm theory of Toeplitz operators with symbols in $ Q$. The essential spectrum of $ {T_f}(f \in Q)$ is shown to be connected and computable in terms of the Stone-Cěch compactification of $ {\mathbf{D}}$. The results in this article partially answer a question posed in [3] and give several new necessary and sufficient conditions for a bounded analytic function on the open unit disc to be in the little Bloch space $ {\mathcal{B}_0}$.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0891638-4
Article copyright: © Copyright 1987 American Mathematical Society

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