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Izv. Akad. Nauk SSSR Ser. Mat., 1971, Volume 35, Issue 4, Pages 940–964 (Mi izv2086)  

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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English version:
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
\Bibitem{GokKru71}
\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
\mathnet{http://mi.mathnet.ru/izv2086}
\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
\issue 4
\pages 955--979
\crossref{https://doi.org/10.1070/IM1971v005n04ABEH001127}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. V. Duduchava, “On bisingular integral operators with discontinuous coefficients”, Math. USSR-Sb., 30:4 (1976), 515–537  mathnet  crossref  mathscinet  zmath  isi
    2. N. L. Vasilevskii, “On a class of singular integral operators with kernels of polar-logarithmic type”, Math. USSR-Izv., 10:1 (1976), 127–143  mathnet  crossref  mathscinet  zmath
    3. Kevin F Clancey, “One dimensional singular integral operators on Lp”, Journal of Mathematical Analysis and Applications, 54:2 (1976), 522  crossref
    4. Martin Costabel, “A singular integral operator related to the one-sided Hilbert transformation”, Integr equ oper theory, 1:2 (1978), 137  crossref  mathscinet  zmath
    5. Martin Costabel, “A contribution to the theory of singular integral equations with carleman shift”, Integr equ oper theory, 2:1 (1979), 11  crossref  mathscinet  zmath
    6. Martin Costabel, “Singular integral operators on curves with corners”, Integr equ oper theory, 3:3 (1980), 323  crossref  mathscinet  zmath
    7. 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  mathnet  crossref  mathscinet  zmath
    8. Yu. I. Karlovich, V. G. Kravchenko, “Systems of singular integral equations with a shift”, Math. USSR-Sb., 44:1 (1983), 75–95  mathnet  crossref  mathscinet  zmath
    9. Bernd Silbermann, “Local objects in the theory of Toeplitz operators”, Integr equ oper theory, 9:5 (1986), 706  crossref  mathscinet  zmath  isi
    10. Roland Duduchava, “On algebras generated by convolutions and discontinuous functions”, Integr equ oper theory, 10:4 (1987), 505  crossref  mathscinet  zmath  isi  elib
    11. Johannes Elschner, “Asymptotics of Solutions to Pseudodifferential Equations of MELLIN Type”, Math Nachr, 130:1 (1987), 267  crossref  mathscinet  zmath  isi
    12. 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  crossref  mathscinet  zmath  isi
    13. 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  crossref  mathscinet  isi  elib
    14. Albrecht Böttcher, Yuri I. Karlovich, “Submultpilicative fuctions and spectral theory of toeplitz operators”, Integral Transforms and Special Functions, 4:1-2 (1996), 181  crossref
    15. A. Yu. Karlovich, “The index of singular integral operators in reflexive Orlicz spaces”, Math. Notes, 64:3 (1998), 330–341  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. C. J. Bishop, A. Böttcher, Yu. I. Karlovich, I. Spitkovsky, “Local Spectra and Index of Singular Integral Operators with Piecewise Continuous Coefficients on Composed Curves”, Math Nachr, 206:1 (1999), 5  crossref  elib
    17. Karlovich Y.I., Lebre A.B., “Algebra of singular integral operators with a Carleman backward slowly oscillating shift”, Integral Equations and Operator Theory, 41:3 (2001), 288–323  crossref  isi  elib
    18. Karlovich Y.I., de A.rellano E.R., “Singular integral operators with fixed singularities on weighted Lebesgue spaces”, Integral Equations and Operator Theory, 48:3 (2004), 331–363  crossref  isi  elib
    19. Karlovich Y.I., “An algebra of pseudodifferential operators with slowly oscillating symbols”, Proceedings of the London Mathematical Society, 92:Part 3 (2006), 713–761  crossref  isi  elib
    20. 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  isi
    21. 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  isi
    22. 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  crossref
    23. A. G. Kamalian, I. M. Spitkovsky, “On the Fredholm Property of a Class of Convolution-Type Operators”, Math. Notes, 104:3 (2018), 404–416  mathnet  crossref  crossref  isi  elib
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