Spectra of partial integral operators with a kernel of three variables

Open Mathematics (2008)

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

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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) = $\underset{a}{\overset{b}{\int }}$ k 1(x, s, y)f(s, y)ds, (T 2 f)(x, y) = $\underset{c}{\overset{d}{\int }}$ 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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