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## Open Mathematics

2008 | 6 | 1 | 149-157
Tytuł artykułu

### Spectra of partial integral operators with a kernel of three variables

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
149-157
Opis fizyczny
Daty
wydano
2008-03-01
online
2008-02-26
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autor
Bibliografia
• [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] Eshkabilov Yu.Kh., On a discrete “three-particle” Schrödinger operator in the Hubbard model, Teoret. Mat. Fiz., 2006, 149, 228–243 (in Russian)
• [3] Eshkabilov Yu.Kh., On the spectrum of tensor sum of the compact operators, Uzbek. Mat. Zh., 2005, 3, 104–112 (in Russian)
• [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] Fenyö S., Beitrag zur Theorie der linearen partiellen Integralgleichungen, Publ. Math. Debrecen, 1955, 4, 98–103
• [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
• [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] 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] 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] 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] Kalitvin A.S., The multispectrum of linear operators, Operators and their applications, Leningrad. Gos. Ped. Inst., Leningrad, 1985, 91–99 (in Russian)
• [12] Kalitvin A.S., Investigations of operators with partial integrals, Candidates Dissertation, Leningrad. Gos. Ped. Inst., 1986 (in Russian)
• [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] 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] 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)
• [16] Lihtarnikov L.M., Vitova L.Z., The solvability of a linear partial integral equation, Ukrain. Mat. Zh., 1976, 28, 83–87 (in Russian)
• [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
• [18] Smirnov V.I., A course in higher mathematics Vol IV Part 1, Nauka, Moscow, 1974 (in Russian)
• [19] Vitova L.Z., On the theory of linear integral equations with partial integrals, Candidates Dissertation, University of Novgorod, 1977 (in Russian)
• [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)
Typ dokumentu
Bibliografia
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