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Sibirsk. Mat. Zh., 2017, Volume 58, Number 5, Pages 1091–1097 (Mi smj2921)  

On systems of linear functional equations of the second kind in $L_2$

V. B. Korotkov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: We consider a general system of functional equations of the second kind in $L_2$ with a continuous linear operator $T$ satisfying the condition that zero lies in the limit spectrum of the adjoint operator $T^*$. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in$L_2$ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in $L_2$ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.

Keywords: system of linear functional equations of the second kind, integral operator, Carleman integral operator, Hilbert–Schmidt operator, Fredholm resolvent, resolvent kernel, spectrum, limit spectrum.

DOI: https://doi.org/10.17377/smzh.2017.58.511

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English version:
Siberian Mathematical Journal, 2017, 58:5, 845–849

Bibliographic databases:

Document Type: Article
UDC: 517.983+517.968.25
MSC: 35R30
Received: 15.11.2016

Citation: V. B. Korotkov, “On systems of linear functional equations of the second kind in $L_2$”, Sibirsk. Mat. Zh., 58:5 (2017), 1091–1097; Siberian Math. J., 58:5 (2017), 845–849

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