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On the factorization of matrix and operator Wiener–Hopf integral equations
N. B. Engibaryan Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan
Abstract:
Let $\widehat{K}$ be a Wiener–Hopf operator,
$\widehat{K}f(x)=\int_0^{\infty}K(x-t)f(t) dt$, $x\ge 0$,
and let $\widehat{K}^*$ be the adjoint operator,
$(f\widehat{K}^*)(t)=\int_0^{\infty}f(x)K(x-t) dx$,
$t\ge 0$, where $K(x)$ belongs to the Banach space
$L_1 (G,(-\infty,\infty))$ of Bochner strongly integrable functions with
values in a Banach algebra $G$. We consider the canonical factorization
problem $I-\widehat{K}=(I-\widehat{V}_-)(I-\widehat{V}_+)$, where $I$ is
the identity operator and $\widehat{V}_-$ (resp. $\widehat{V}_+ $) is
a left (resp. right) triangular convolution operator such that the operators
$I-\widehat{V}_{\pm}$ are invertible in the spaces $L_{p} (G,(0,\infty))$,
$1\le p\le \infty$. We put forward a semi-inverse factorization method and prove
that the canonical factorization exists if and only if the operators
$I-\widehat{K}$ and $I-\widehat{K}^*$ are invertible
in $L_1 (G,(0,\infty))$.
Keywords:
operator Wiener–Hopf integral equation, strongly integrable functions,
semi-inverse Volterra factorization method.
DOI:
https://doi.org/10.4213/im8584
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English version:
Izvestiya: Mathematics, 2018, 82:2, 273–282
Bibliographic databases:
UDC:
517.968.25+517.968.28
MSC: 45E10, 45F15, 47B35 Received: 16.06.2016
Citation:
N. B. Engibaryan, “On the factorization of matrix and operator Wiener–Hopf integral equations”, Izv. RAN. Ser. Mat., 82:2 (2018), 33–42; Izv. Math., 82:2 (2018), 273–282
Citation in format AMSBIB
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