Abstract
We study the algebra of operators with the Bergman kernel extended by isometric weighted shift operators. The coefficients of the algebra are assumed to be automorphic with respect to a cyclic parabolic group of fractional-linear transformations of a unit disk and continuous on the Riemann surface of the group. By using an isometric transformation, we obtain a quasiautomorphic matrix operator on the Riemann surface with properties similar to the properties of the Bergman operator. This enables us to construct the algebra of symbols, devise an efficient criterion for the Fredholm property, and calculate the index of the operators of the algebra considered.
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REFERENCES
A. D. Dzhuraev, Method of Singular Integral Equations [in Russian], Nauka, Moscow (1984).
N. L. Vasilevskii, Multidimensional Singular Integral Operators with Discontinuous Classical Symbols [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Odessa (1985).
Yu. I. Karlovich, Algebras of Operators of Convolution Type with Discrete Groups of Shifts and Oscillation Coefficients [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Odessa (1990).
A. B. Antonevich, Linear Functional Equations: Operator Approach [in Russian], Universitetskoe, Minsk (1988).
G. Dzhangibekov, “On the algebra generated by polykernel operators with shift,” Dokl. Akad. Nauk Tadzh. SSR, 34, No. 7, 399–403 (1991).
R. Duduchava, A. Saginashvili, and E. Shargorodsky, “On two-dimensional singular integral operators with Carleman shift,” J. Oper. Theory, 37, 263–279 (1997).
A. F. Beardon, The Geometry of Discrete Groups [Russian translation], Nauka, Moscow (1986).
G. Springer, Introduction to Riemann Surfaces, Addison-Wesley, New York (1957).
N. Ya. Krupnik, “On the exact constant in the Simonenko theorem on the envelope of a family of operators of local type,” Funkts. Anal. Prilozhen., 20, No. 2, 70–72 (1986).
I. Ts. Gokhberg and I. A. Fel'dman, Convolution Equations and Projection Methods for Their Solution [in Russian], Nauka, Moscow (1971).
M. A. Naimark, Normed Rings [in Russian], Nauka, Moscow (1968).
S. Bochner and R. S. Phillips, “Absolutely convergent Fourier expansion for noncommutative normed rings,” Ann. Math., 43, No. 2, 409–418 (1942).
B. V. Fedosov, “On the index of an elliptic system on a manifold,” Funkts. Anal. Prilozhen., 4, No. 4, 57–67 (1970).
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Mozel', V.A., Chernetskii, V.A. Algebra of Bergman Operators with Automorphic Coefficients and Parabolic Group of Shifts. Ukrainian Mathematical Journal 53, 1464–1472 (2001). https://doi.org/10.1023/A:1014362524114
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DOI: https://doi.org/10.1023/A:1014362524114