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

This article is cited in 7 scientific papers (total in 7 papers)

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

Full text: PDF file (283 kB)
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English version:
Sbornik: Mathematics, 2000, 191:12, 1809–1825

Bibliographic databases:

UDC: 517.9
MSC: 45B05, 45D05, 47Gxx
Received: 27.04.1999

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
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\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
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\transl
\jour Sb. Math.
\yr 2000
\vol 191
\issue 12
\pages 1809--1825
\crossref{https://doi.org/10.1070/sm2000v191n12ABEH000529}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0034340591}


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  • http://mi.mathnet.ru/eng/msb/v191/i12/p61

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    2. Yengibarian N.B., “Factorization of Markov chains”, J. Theoret. Probab., 17:2 (2004), 459–481  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. N. B. Engibaryan, “On the fixed points of monotonic operators in the critical case”, Izv. Math., 70:5 (2006), 931–947  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. N. B. Engibaryan, “On the factorization of integral operators on spaces of summable functions”, Izv. Math., 73:5 (2009), 921–937  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. N. B. Engibaryan, “Differential equations where the derivative is taken with respect to a measure”, Sb. Math., 202:2 (2011), 243–256  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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