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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.
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
Linking options:
https://www.mathnet.ru/eng/smj2921 https://www.mathnet.ru/eng/smj/v58/i5/p1091
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Abstract page: | 167 | Full-text PDF : | 31 | References: | 29 | First page: | 5 |
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