Compact Toeplitz operators on Bergman spaces (Q1962716)

Let \(\varphi\) be a bounded measurable function on the unit disc \(\mathbb{D}\) and \(T_\varphi\) the corresponding Toeplitz operator on the Bergman space of square-integrable holomorphic functions on \(\mathbb{D}\). For \(\varphi\) radial (i.e. satisfying \(\varphi(z)=\varphi(|z|)\)), it was shown by \textit{B. Korenblum} and \textit{K. Zhu} [J. Oper. Theory 33, 353-361 (1995; Zbl 0837.47022)] that \(T_\varphi\) is compact if and only if \(\lim_{x\to 1} \int_x^1 \varphi(r) dr=0\) (*). In this paper, the author gives a number of necessary or sufficient conditions for the compactness of \(T_\varphi\) for not necessarily radial symbols \(\varphi\). Typically, these involve the analogs of the condition (*) for the Fourier components \(\varphi_j\) of \(\varphi\), or some additional hypothesis on \(\varphi\) like nonnegativity, continuity of the map \(r\mapsto\varphi(r e^{i\theta})\) from \((0,1)\) into \(L^\infty(\partial\mathbb{D})\), or of \(\theta\mapsto \varphi(re^{i\theta})\) from \(\partial\mathbb{D}\) into \(L^\infty(0,1)\). (Remark: In part (6) of Theorem~1, the assumption ``\(\varphi\) is real'' should read ``\(\varphi\) is nonnegative''.) A necessary and sufficient criterion for compactness of \(T_\varphi\) has recently been given by \textit{S. Axler} and \textit{D. Zheng} [Indiana Univ. Math. J. 47, 387-400 (1998; Zbl 0914.47029)]: for \(\varphi\) bounded measurable, \(T_\varphi\) is compact if and only if \(\lim_{|x|\nearrow 1} (1-|x|^2) ^2\int_{\mathbb{D}} \varphi(y) |1-\overline xy|^{-4} dm(y)=0\), where \(dm\) is the Lebesgue area measure on~\(\mathbb{D}\).