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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 2, Pages 33–42 (Mi izv8584)  

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.

Funding Agency Grant Number
State Committee on Science of the Ministry of Education and Science of the Republic of Armenia 15T-1A246
This investigation has been carried out with the financial support of the State Committee on Science MSE RA under scientific project no. 15T-1A246.


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

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