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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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VMO, ESV, and Toeplitz operators on the Bergman space
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by Ke He Zhu PDF
Trans. Amer. Math. Soc. 302 (1987), 617-646 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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