Examples of compact Toeplitz operators on the Bergman space (Q5957963)

The author gives some counterexamples to a question on compactness of Toeplitz operators in Bergman spaces on the unit disc, formulated by \textit{R. Yoneda} [Hokkaido Math. J. 28, 563-576 (1999; Zbl 0944.47018)]. Namely, let \(E_n=[a_n,a_{n+1})\) be a sequence of successive intervals on \([0,1], \;\;0=a_0<a_1< \cdots < a_n\), where \(n \to 1\) as \(n\to\infty\). The question is whether the Toeplitz operator NEWLINE\[NEWLINET_\phi \quad \text{with} \quad \phi(re^{i\theta})=\sum_{n=0}^\infty e^{in\theta} \chi_{E_n}(r)NEWLINE\]NEWLINE is compact or not? The author gives examples which show that both cases occur.