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
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.
To Professor Roland Duduchava on the occasion of his 70th birthday
Mathematics Subject Classification (2010). Primary 47L15; Secondary 47G10, 47L30.
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Espinoza-Loyola, E., Karlovich, Y.I. (2017). C *-algebras of Bergman Type Operators with Piecewise Continuous Coefficients over Bounded Polygonal Domains. In: Maz'ya, V., Natroshvili, D., Shargorodsky, E., Wendland, W. (eds) Recent Trends in Operator Theory and Partial Differential Equations. Operator Theory: Advances and Applications, vol 258. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-47079-5_8
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DOI: https://doi.org/10.1007/978-3-319-47079-5_8
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Publisher Name: Birkhäuser, Cham
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