Extension of Zhu's solution to Lotto's conjecture on the weighted Bergman spaces (Q1777681)

Let \(D\) denote the unit disc in \(\mathbb C\), \(\, H\) the space of all analytic functions on \(D\), \(\, A_\alpha^2\) the weighted Bergman space \((\alpha > -1)\), \(\, H^2\) the Hardy space and \({\mathcal L}_p(X)\) the Schatten \(p\)-class of operators on \(X\) (\(X = H^2\) or \(X = A_\alpha^2\)), \(0 < p < \infty\). In [Contemp.\ Math.\ 213, 93--97 (1998; Zbl 0898.47025)], \textit{B.~A.\ Lotto} conjectured the existence of a ``simple'' domain \(G_p\) such that the Riemann map from \(D\) onto \(G_p\) induces a compact operator which does not belong to \({\mathcal L}_p (H^2)\). This conjecture was proved by \textit{Y.--S.\ Zhu} [Int.\ J.\ Math.\ Math.\ Sci.\ 26, No.~4, 239--248 (2001; Zbl 0996.47034)]. The paper under the review is devoted to an extension of Zhu's construction to prove the same conjecture in the setting of \({\mathcal L}_p(A_\alpha^2)\)-spaces.