VMO, ESV, and Toeplitz operators on the Bergman space

Zhu, Ke He

doi:10.1090/S0002-9947-1987-0891638-4

VMO, ESV, and Toeplitz operators on the Bergman space
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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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