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This article is cited in 23 scientific papers (total in 23 papers)
Singular integral operators with piecewise continuous coefficients and their symbols
I. Ts. Gokhberg, N. Ya. Krupnik
Abstract:
The algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients is studied. Operators acting on the space $L_p$ with a weight are considered. The contour is assumed to consist of closed and open arcs. The structure of the symbols of the operators considered is elucidated. It is found that the symbol is a matrix-function of the second order depending both on $p$ and on the weight. Criteria that the operator be Fredholm and a formula for its index are established.
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Mathematics of the USSR-Izvestiya, 1971, 5:4, 955–979
Bibliographic databases:
 
UDC:
517.94
MSC: Primary 45E05, 45F05; Secondary 46L20, 47B30, 47G05
Received: 16.06.1970
Citation:
I. Ts. Gokhberg, N. Ya. Krupnik, “Singular integral operators with piecewise continuous coefficients and their symbols”, Izv. Akad. Nauk SSSR Ser. Mat., 35:4 (1971), 940–964; Math. USSR-Izv., 5:4 (1971), 955–979
Citation in format AMSBIB
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\by I.~Ts.~Gokhberg, N.~Ya.~Krupnik
\paper Singular integral operators with piecewise continuous coefficients and their symbols
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1971
\vol 35
\issue 4
\pages 940--964
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=291893}
\zmath{https://zbmath.org/?q=an:0235.47025}
\transl
\jour Math. USSR-Izv.
\yr 1971
\vol 5
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\pages 955--979
\crossref{https://doi.org/10.1070/IM1971v005n04ABEH001127}
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Martin Costabel, “A singular integral operator related to the one-sided Hilbert transformation”, Integr equ oper theory, 1:2 (1978), 137
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Martin Costabel, “Singular integral operators on curves with corners”, Integr equ oper theory, 3:3 (1980), 323
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N. Ya. Krupnik, “A sufficient set of $n$-dimensional representations of a Banach algebra and the $n$-symbol”, Funct. Anal. Appl., 14:1 (1980), 50–52
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Yu. I. Karlovich, V. G. Kravchenko, “Systems of singular integral equations with a shift”, Math. USSR-Sb., 44:1 (1983), 75–95
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Bernd Silbermann, “Local objects in the theory of Toeplitz operators”, Integr equ oper theory, 9:5 (1986), 706
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Roland Duduchava, “On algebras generated by convolutions and discontinuous functions”, Integr equ oper theory, 10:4 (1987), 505
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Johannes Elschner, “Asymptotics of Solutions to Pseudodifferential Equations of MELLIN Type”, Math Nachr, 130:1 (1987), 267
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R. Duduchava, T. Latsabidze, A. Saginashvili, “Singular integral operators with the complex conjugation on curves with cusps”, Integr equ oper theory, 22:1 (1995), 1
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A. Böttcher, Yu. I. Karlovich, V. S. Rabinovich, “Emergence, persistence, and disappearance of logarithmic spirals in the spectra of singular integral operators”, Integr equ oper theory, 25:4 (1996), 406
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A. Yu. Karlovich, “The index of singular integral operators in reflexive Orlicz spaces”, Math. Notes, 64:3 (1998), 330–341
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Karlovich Y.I., “An algebra of pseudodifferential operators with slowly oscillating symbols”, Proceedings of the London Mathematical Society, 92:Part 3 (2006), 713–761
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Karlovich A.Yu., “Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights”, Operator Algebras, Operator Theory and Applications, Operator Theory Advances and Applications, 195, 2010, 185–212
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Karlovich A.Yu., “Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves”, Topics in Operator Theory: Operators, Matrices and Analytic Functions, Operator Theory Advances and Applications, 1, 2010, 321–336
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Martin Costabel, “An inverse for the Gohberg-Krupnik symbol map”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 87:1-2 (2011), 153
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A. G. Kamalian, I. M. Spitkovsky, “On the Fredholm Property of a Class of Convolution-Type Operators”, Math. Notes, 104:3 (2018), 404–416
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