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On the factorization of matrix and operator Wiener-Hopf integral equations. (English. Russian original) Zbl 1395.45006

Izv. Math. 82, No. 2, 273-282 (2018); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 82, No. 2, 33-42 (2018).
The paper considers the Wiener-Hopf operator \((\hat{K}f)(x)=\int _{0}^{\infty }k(x-t)f(t)dt,\, x\geq 0 \), 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. The autor considers the canonical factorization problem \(I-\hat{K}=(I-\hat{V}_{-} )(I-\hat{V}_{+} )\), where \(I\) is the identity operator and \(\hat{V}_{-} \) (resp. \(\hat{V}_{+} \)) is a left (resp. right) triangular convolution operator such that the operators \(I-\hat{V}_{\pm } \) are invertible in the spaces \(L_{p} (G,(0,+\infty )),\; 1\leq p\leq +\infty \). The author puts forward a semi-inverse factorization method and prove that the canonical factorization exists if and only if the operators \(I-\hat{K},\; I-\hat{K}^{*} \) (\(\hat{K}^{*}\)-adjoint operator) are invertible in \(L_{1} (G,(0,+\infty ))\).
It is clear that the obtained criterion of invertibility in terms of factorization is not effective, but in the situation under consideration, it is unlikely to succeed in obtaining more effective criterion.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45F15 Systems of singular linear integral equations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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References:

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