Algebras generated by the Bergman and anti-Bergman projections and by multiplications by piecewise continuous functions
Let \(\Pi\) be the open upper half-plane of the complex plane \(\mathbb C\) equipped with the Lebesgue area measure \(dA(w)=dx\,dy\), and let \(B_{\Pi}\) and \(\widetilde{B}_{\Pi}\) denote, respectively, the \textit{Bergman} and \textit{anti-Bergman} projections in \(L^{2}(\Pi)\), given by the formulas \[ (B_{\Pi}f)(z) = -\frac{1}{\pi} \int_{\Pi} \frac{f(w)}{(z-\bar{w})^{2}}\,dA(w), \qquad f\in L^{2}(\Pi), \qquad z\in\Pi, \] \[ (\widetilde{B}_{\Pi}f)(z) = -\frac{1}{\pi} \int_{\Pi} \frac{f(w)}{(\bar{z}-w)^{2}}\,dA(w), \qquad f\in L^{2}(\Pi), \qquad z\in\Pi. \] Denote by \(\dot{\overline{\Pi}}\) the one-point compactification of the closed upper half-plane in \(\mathbb C\). Let \(\mathcal L\) be a finite union of Lyapunov curves in \(\dot{\overline{\Pi}}\) and let \(PC(\mathcal L)\) be the \(C^{\ast}\)-subalgebra of \(L^{\infty}(\Pi)\) consisting of all continuous functions on \(\dot{\overline{\Pi}}\setminus \mathcal L\) which have the one-sided limits at the points of \(\mathcal L\). Denote by \(\mathfrak A\) the \(C^{\ast}\) algebra of bounded linear operators on the space \(L^{2}(\Pi)\) generated by \(B_{\Pi}\), \(\widetilde{B}_{\Pi}\) and by the operators of multiplication by the functions in \(PC(\mathcal L)\). The authors construct a symbol calculus for \(\mathfrak A\) and obtain a Fredholm criterion for operators \(A\in \mathfrak A\). They use a local principle, limit operators techniques, and Plamenevskij's results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities. It is also worth to mention an auxiliary result: a symbolic calculus for the unital \(C^{\ast}\)-algebra generated by a finite set of orthogonal projections, the sum of which equals the unit and a finite set of orthogonal one-dimensional projections.
scientific article; zbMATH DE number 5003806