C *-algebras of Bergman Type Operators with Piecewise Continuous Coefficients over Bounded Polygonal Domains

Abstract

Given \(\alpha\;\in\;(0,2]\), we study the C * -algebra \(\mathfrak{A}_{\mathbb{K}_{a}}\) generated by the operators of multiplication by piecewise constant functions with discontinuities on a finite union \(\mathcal{L}_\omega\) of rays starting from the origin and by the Bergman and anti-Bergman projections acting on the Lebesgue space \(L^{2}(\mathbb{K}_\alpha)\) over the open sector $$ \mathbb{K}_\alpha = \{{z=re^{i\theta}:r>0, \theta\in (0,\pi\alpha)} \}.$$ Then, for any bounded simply connected polygonal domain U, the C *-algebra \(\mathfrak{B}_U\) generated by the operators of multiplication by piecewise continuous functions with discontinuities on a finite union \(\mathcal{L}\subset U\) of straight line segments and by the Bergman and anti-Bergman projections acting on the Lebesgue space \(L^{2}(U)\) is investigated. Symbol calculi for the C *-algebra \(\mathfrak{A}_{\mathbb{K}_{a}} \mathrm{and}\; \mathfrak{B}_U\) are constructed and an invertibility criterion for the operators \(A\;\in\;\mathfrak{A}_{\mathbb{K}_{a}}\) and a Fredholm criterion for the operators \(A\;\in\;\mathfrak{B}_U\) in terms of their symbols are established.