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Partially integral operators with bounded kernels

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Abstract

Let Ω = [a, b]ν and let T be a partially integral operator defined in L 2(Ω2) as follows:

$$ (Tf)(x,y) = \int_\Omega {q(x,s,y)f(s,y)} d\mu (s). $$

In the article, we study the solvability of the partially integral Fredholm equations f − ℵTf = g, where g ∈ L 2(Ω2) is a given function and ℵ ∈ ℂ. The notion of determinant (which is a measurable function on Ω) is introduced for the operator E − ℵT, with E is the identity operator in L 2(Ω2). Some theorems on the spectrum of a bounded operator T are proven.

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References

  1. J. Appell J., E. V. Frolova, A. S. Kalitvin, and P. P. Zabrejko, “Partial integral operators on C([a, b] × [c, d]),” Integral Equations Oper. Theory 27(2), 125–140 (1997).

    Article  MATH  Google Scholar 

  2. T. Carleman, “Zur Theorie der linearen Integralgleichungen,” Math. Z. 9(3),196–217 (1921).

    Article  MathSciNet  Google Scholar 

  3. Yu. Kh. Eshkabilov, “On a discrete “three-particle” Schroedinger operator in the Hubbard model,” Teor.Mat. Fiz. 149(2), 228–243 (2006) [Theor.Math. Phys. 149 (2), 1497–1511 (2006)].

    MathSciNet  Google Scholar 

  4. K. O. Friedrichs, Perturbation of Spectra in Hilbert Space (Amer. Math. Soc., Providence, Rhode Island, 1965).

    MATH  Google Scholar 

  5. V. A. Kakichev and N. V. Kovalenko, “Towards a theory of two-dimensional integral equations with partial integrals,” Ukr. Mat. Zh. 25(3), 302–312 (1973) [Ukr.Math. J. 25 (3), 236–245 (1973)].

    MATH  Google Scholar 

  6. A. S. Kalitvin and P. P. Zabrejko, “On the theory of partial integral operators,” J. Integral Equations Appl. 3(3), 351–382 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Fifth edition. With a supplement “Banach algebras” by V. M. Tikhomirov. Nauka, Moscow, 1981) [Dover Publications, 1999].

    MATH  Google Scholar 

  8. K. K. Kudaybergenov, “∇-Fredholm operators in Banach-Kantorovich spaces,” Methods Funct. Anal. Topol. 12(3), 234–242 (2006).

    MATH  MathSciNet  Google Scholar 

  9. D. Mattis, “The few-body problem in a lattice,” Rev. Modern Phys. 58(2), 361–379 (1986).

    Article  MathSciNet  Google Scholar 

  10. S. G. Mikhlin, “On convergence of Fredholm’s series,” Dokl. Acad. Nauk. SSSR 42(9), 373–376 (1944).

    Google Scholar 

  11. S. G. Mikhlin, Linear Integral Equations (Fizmatgiz, Moscow, 1959) [Hindustan Publ.Corp.,Delhi, 1960].

    Google Scholar 

  12. A. I. Mogilner, “Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: problems and results,” Adv. Sov. Math. 5, 139–194 (1991).

    MathSciNet  Google Scholar 

  13. V. A. Morozov, “Application of the method of regularization to the solution of an ill-posed problem,” Vestn. Mosk. Univ., Ser. I Mat. Meh. (4), 13–21 (1965).

  14. O. P. Okolelov, “Analogs of some Fredholm’s theorems for integral equations with multiple and partial integrals,” Trudy Irkutsk. Gos. Univ. 26, 74–89 (1968).

    MathSciNet  Google Scholar 

  15. V. I. Smirnov, A Course in Higher Mathematics, vol. IV, part 1 (Sixth edition, Nauka, Moscow, 1974) [Pergamon Press, 1964].

    Google Scholar 

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Authors and Affiliations

  1. National University of Uzbekistan, Tashkent, 100174, Uzbekistan

    Yu. Kh. Eshkabilov

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  1. Yu. Kh. Eshkabilov
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Correspondence to Yu. Kh. Eshkabilov.

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Original Russian Text © Yu. Kh. Eshkabilov, 2008, published in Matematicheskie Trudy, 2008, Vol. 11, No. 1, pp. 192–207.

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Eshkabilov, Y.K. Partially integral operators with bounded kernels. Sib. Adv. Math. 19, 151–161 (2009). https://doi.org/10.3103/S1055134409030018

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  • DOI: https://doi.org/10.3103/S1055134409030018

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