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 Mat. Sb., 2000, Volume 191, Number 12, Pages 61–76 (Mi msb529)

Setting and solving several factorization problems for integral operators

N. B. Engibaryan

Byurakan Astrophysical Observatory, National Academy of Sciences of Armenia

Abstract: The problem of factorization
$$I-K=(I-U_-)(I-U_+),$$
is considered. Here $I$ is the identity operator, $K$ is a fixed integral operator of Fredholm type:
$$(Kf)(x)=\int_a^bk(x,t)f(t) dt, \qquad -\infty\leqslant a<b\leqslant+\infty,$$
$U_\pm$ are unknown upper and lower Volterra operators. Classes of generalized Volterra operators $U_\pm$ are introduced such that $I-U_\pm$ are not necessarily invertible operators in the spaces of functions on $(a,b)$ under consideration. A combination of the method of non-linear factorization equations and a priori estimates brings forth new results on the existence and properties of the solution to this problem for $k\geqslant 0$, both in the subcritical case $\mu<1$ and in the critical case $\mu=1$, where $\mu=r(K)$ the spectral radius of the operator $K$. In addition, the problem of non-Volterra factorization is posed and studied, when the kernels of $U_+$ and $U_-$ vanish on some parts $S_-$ and $S_+$ of the domain $S=(a,b)^2$ such that $S_+\cup S_-=S$.

DOI: https://doi.org/10.4213/sm529

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English version:
Sbornik: Mathematics, 2000, 191:12, 1809–1825

Bibliographic databases:

UDC: 517.9
MSC: 45B05, 45D05, 47Gxx

Citation: N. B. Engibaryan, “Setting and solving several factorization problems for integral operators”, Mat. Sb., 191:12 (2000), 61–76; Sb. Math., 191:12 (2000), 1809–1825

Citation in format AMSBIB
\Bibitem{Eng00} \by N.~B.~Engibaryan \paper Setting and solving several factorization problems for integral operators \jour Mat. Sb. \yr 2000 \vol 191 \issue 12 \pages 61--76 \mathnet{http://mi.mathnet.ru/msb529} \crossref{https://doi.org/10.4213/sm529} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1829414} \zmath{https://zbmath.org/?q=an:1009.45013} \transl \jour Sb. Math. \yr 2000 \vol 191 \issue 12 \pages 1809--1825 \crossref{https://doi.org/10.1070/sm2000v191n12ABEH000529} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000168023700010} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0034340591} 

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• https://doi.org/10.4213/sm529
• http://mi.mathnet.ru/eng/msb/v191/i12/p61

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Grigoryan G.A., “Special factorization of a noninvertible Fredholm operator of the second kind”, Differ. Equ., 38:12 (2002), 1792–1800
2. Yengibarian N.B., “Factorization of Markov chains”, J. Theoret. Probab., 17:2 (2004), 459–481
3. N. B. Engibaryan, “On the fixed points of monotonic operators in the critical case”, Izv. Math., 70:5 (2006), 931–947
4. G. A. Grigoryan, “Special factorization of a non-invertible integral Fredholm operator of the second kind with Hilbert–Schmidt kernel”, Sb. Math., 198:5 (2007), 627–637
5. N. B. Engibaryan, “On the factorization of integral operators on spaces of summable functions”, Izv. Math., 73:5 (2009), 921–937
6. N. B. Engibaryan, “Differential equations where the derivative is taken with respect to a measure”, Sb. Math., 202:2 (2011), 243–256
7. G. A. Grigoryan, “On a Criterion for the Invertibility of Integral Operators of the Second Kind in the Space of Summable Functions on the Semiaxis”, Math. Notes, 96:6 (2014), 914–920
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