The density problem for self-commutators of unbounded Bergman operators (Q1780973)

If \(G\) is an open subset of the complex plane \(\mathbb C\), let \(\text{Hol}\,(G)\) denote the class of analytic functions on \(G\), and let \(H^\infty(G)\) denote the class of bounded analytic functions on \(G\). If \(\varphi\in\text{Hol}\,(G)\), then the multiplication operator \(M_{\varphi,G}\) is defined by \(M_{\varphi,G}(f)=\varphi f\). The unbounded Bergman operator \(M_{\theta,G}\), with \(\theta(z)= z\), is denoted by \(S\). In the main results of this paper, the authors show that (1) if \(G\) is a simply connected region of finite area, then the Bergman operator \(S\) and its self commutator \([S^*,S]= S^*S- SS^*\) are densely defined; (2) if \(G\) is an open subset of the plane which lies entirely within a sector of angle \({1\over 2}\pi\), then the Bergman operator \(S\) and its self-commutator \(S^*S- SS^*\) are densely defined.