Spectra of partial integral operators with a kernel of three variables

Yusup Eshkabilov

Open Mathematics (2008)

  • Volume: 6, Issue: 1, page 149-157
  • ISSN: 2391-5455

Abstract

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Let Ω= [a, b] × [c, d] and T 1, T 2 be partial integral operators in C (Ω): (T 1 f)(x, y) = ∫ a b k 1(x, s, y)f(s, y)ds, (T 2 f)(x, y) = ∫ c d k 2(x, ts, y)f(t, y)dt where k 1 and k 2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT 1, τ ∈ ℂ and E−τT 2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T 1, T 2, and T = T 1 + T 2 are proved.

How to cite

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Yusup Eshkabilov. "Spectra of partial integral operators with a kernel of three variables." Open Mathematics 6.1 (2008): 149-157. <http://eudml.org/doc/269490>.

@article{YusupEshkabilov2008,
abstract = {Let Ω= [a, b] × [c, d] and T 1, T 2 be partial integral operators in \[ C \] (Ω): (T 1 f)(x, y) = \[ \mathop \smallint \limits \_a^b \] k 1(x, s, y)f(s, y)ds, (T 2 f)(x, y) = \[ \mathop \smallint \limits \_c^d \] k 2(x, ts, y)f(t, y)dt where k 1 and k 2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT 1, τ ∈ ℂ and E−τT 2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T 1, T 2, and T = T 1 + T 2 are proved.},
author = {Yusup Eshkabilov},
journal = {Open Mathematics},
keywords = {partial integral operator; partial integral equation; Fredholm integral equation; Fredholm determinant; Fredholm minor; spectrum; limit spectrum; point spectrum},
language = {eng},
number = {1},
pages = {149-157},
title = {Spectra of partial integral operators with a kernel of three variables},
url = {http://eudml.org/doc/269490},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Yusup Eshkabilov
TI - Spectra of partial integral operators with a kernel of three variables
JO - Open Mathematics
PY - 2008
VL - 6
IS - 1
SP - 149
EP - 157
AB - Let Ω= [a, b] × [c, d] and T 1, T 2 be partial integral operators in \[ C \] (Ω): (T 1 f)(x, y) = \[ \mathop \smallint \limits _a^b \] k 1(x, s, y)f(s, y)ds, (T 2 f)(x, y) = \[ \mathop \smallint \limits _c^d \] k 2(x, ts, y)f(t, y)dt where k 1 and k 2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT 1, τ ∈ ℂ and E−τT 2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T 1, T 2, and T = T 1 + T 2 are proved.
LA - eng
KW - partial integral operator; partial integral equation; Fredholm integral equation; Fredholm determinant; Fredholm minor; spectrum; limit spectrum; point spectrum
UR - http://eudml.org/doc/269490
ER -

References

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  1. [1] Appell J., Kalitvin A.S., Nashed M.Z., On some partial integral equations arising in the mechanics of solids, ZAMM Z. Angew. Math. Mech., 1999, 79, 703–713 http://dx.doi.org/10.1002/(SICI)1521-4001(199910)79:10<703::AID-ZAMM703>3.0.CO;2-W 
  2. [2] Eshkabilov Yu.Kh., On a discrete “three-particle” Schrödinger operator in the Hubbard model, Teoret. Mat. Fiz., 2006, 149, 228–243 (in Russian) 
  3. [3] Eshkabilov Yu.Kh., On the spectrum of tensor sum of the compact operators, Uzbek. Mat. Zh., 2005, 3, 104–112 (in Russian) 
  4. [4] Eshkabilov Yu.Kh., Perturbation of spectra of the operator multiplication with PIO, Acta of National University of Uzbekistan, Tashkent, 2006, 2, 17–21 (in Russian) 
  5. [5] Fenyö S., Beitrag zur Theorie der linearen partiellen Integralgleichungen, Publ. Math. Debrecen, 1955, 4, 98–103 Zbl0064.10101
  6. [6] Friedrichs K.O., Perturbation of spectra in Hilbert space, In: Kac M. (Ed.), Lectures in Applied Mathematics, Proceedings of the Summer Seminar (1960 Boulder Colorado), Vol.III American Mathematical Society, Providence, R.I. 1965 Zbl0142.11001
  7. [7] Kakichev V.A., Kovalenko N.V., On the theory of two-dimensional partial integral equations, Ukrain. Mat. Zh., 1973, 25, 302–312 (in Russian) 
  8. [8] Kalitvin A.S., On the solvability of some classes of integral equations with partial integrals, Functional analysis (Russian), Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1989, 29, 68–73 (in Russian) 
  9. [9] Kalitvin A.S., On the spectrum and eigenfunctions of operators with partial integrals of V.I. Romanovskiĭtype, Functional analysis (Russian), Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1984, 22, 35–45 (in Russian) 
  10. [10] Kalitvin A.S., The spectrum of some classes of operators with partial integrals, Operators and their applications, Leningrad. Gos. Ped. Inst., Leningrad, 1985, 27–35 (in Russian) 
  11. [11] Kalitvin A.S., The multispectrum of linear operators, Operators and their applications, Leningrad. Gos. Ped. Inst., Leningrad, 1985, 91–99 (in Russian) 
  12. [12] Kalitvin A.S., Investigations of operators with partial integrals, Candidates Dissertation, Leningrad. Gos. Ped. Inst., 1986 (in Russian) 
  13. [13] Kalitvin A.S., The spectrum of linear operators with partial integrals and positive kernels, Operators and their applications, Leningrad. Gos. Ped. Inst., Leningrad, 1988, 43–50 (in Russian) 
  14. [14] Lihtarnikov L.M., Spevak L.V., A linear partial integral equation of V.I. Romanovskiĭ type with two parameters, Differencialnye Uravnenija, 1976, 7, 165–176 (in Russian) 
  15. [15] Lihtarnikov L.M., Spevak L.V., The solvability of a linear integral equation of V. I. Romanovskiĭ type with partial integrals, Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1976, 7, 106–115 (in Russian) Zbl0436.45024
  16. [16] Lihtarnikov L.M., Vitova L.Z., The solvability of a linear partial integral equation, Ukrain. Mat. Zh., 1976, 28, 83–87 (in Russian) Zbl0333.45002
  17. [17] Mogilner A.I., Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: problems and results, Adv. Soviet Math., 1991, 5, 139–194, Amer. Math. Soc., Providence, RI, 1991 Zbl0741.34055
  18. [18] Smirnov V.I., A course in higher mathematics Vol IV Part 1, Nauka, Moscow, 1974 (in Russian) 
  19. [19] Vitova L.Z., On the theory of linear integral equations with partial integrals, Candidates Dissertation, University of Novgorod, 1977 (in Russian) 
  20. [20] Vitova L.Z., Solvability of a partial integral equation with degenerate kernels, Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1976, 7, 41–52 (in Russian) Zbl0456.47041

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