Toeplitz operators and the essential boundary on polyanalytic functions (Q2842040)

Let \(U\) be a bounded domain (that is, an open and connected set) in the complex plane \(\mathbb C\). For a positive (negative) integer \(j\), the poly-Bergman space \({\mathcal A}_j^2(U)\) is defined as the subspace of the Lebesgue space \(L^2(U,dA)\) consisting of infinitely differentiable functions \(f\) on \(U\) satisfying \(\partial^j_{\overline{z}}f=0\) (respectively, \(\partial^{-j}_{z}f=0\)). The main result of the paper is the Fredholm symbol calculus for the \(C^*\)-algebra generated by the poly-Bergman projections \(B_{U,j}\) of order \(j\) (that is, the orthogonal projection of \(L^2(U,dA)\) onto \(A_j^2(U)\)) and the multiplication operators by all the functions continuous on the closure of \(U\). It is based on the local nature of the poly-Bergman projections for \(j\neq 0\) (also established in the paper) and stated in terms of the \(j\)-removable boundary \(\partial_r^jU\) of \(U\), defined as the subset of the ``regular'' boundary \(\partial U\) at the points of which \(B_{U,j}\) is locally equivalent to zero. It is shown that \(\partial_r^jU\) is nothing but the set of all isolated points of \(\partial U\) if \(j\neq\pm 1\), while \(\partial_r^{\pm 1}U\) coincides with the set \(\partial_{2-r}U\) introduced in [\textit{S. Axler, J. Conway} and \textit{G. McDonald}, Can. J. Math. 34, 466--483 (1982; Zbl 0452.47032)]. Also, the structure of \({\mathcal A}_j^2(U)\ominus{\mathcal A}_j^2(U^j_\infty)\) is described, where \(U^j_\infty\) is the union of \(U\) with the Cantor-Bendixson derivative of \(\partial U\).