Comments to the problems of small perturbations of linear equations and linear term of the spectral characteristics of a Fredholm operator
A. A. Kyashkin1, B. V. Loginov2, P. A. Shamanaev3
| Annotation | In the monograph \cite{kyashkinb1} and the article \cite{kyashkinb2} the problem on perturbation of linear equation by small linear summand of the form (B−εA)x=h were investigated with closely defined on DB Fredholmian operator B:E1⊃DB→E2, DA⊃DB, or A∈L{E1,E2}, ε∈C1 - small parameter, E1 and E2 - are Banach spaces. The application of the results \citetwo{kyashkinb3}{kyashkinb4} formulated in the form of the lemma on the biorthogonality of generalized Jordan chains allows to give some retainings of the results \citetwo{kyashkinb1}{kyashkinb2}. This problem is considered here in the general case of sufficiently smooth (analytic) by ε operator-function A(ε). It is given also the application of the biorthogonality lemma and branching equation in the root subspaces to the problem on perturbation of Fredholm points in C-spectrum of the operator A(0). |
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| Keywords | linear operators in Banach spaces, perturbation theory |
1Graduate student of chair of an applied mathematics, Mordovian State University of a name of linebreak N. P. Ogarev, Saransk; andrej_kjashkin@list.ru.%
2Professor department of Mathematics, Ulyanovsk State Technical University, Ulyanovsk; loginov@ulstu.ru%
3Head of Applied Mathematics Chair,; Mordovian State University after N. P. Ogarev,; Saransk;linebreak korspa@yandex.ru.%
Citation: A. A. Kyashkin, B. V. Loginov, P. A. Shamanaev, "[Comments to the problems of small perturbations of linear equations and linear term of the spectral characteristics of a Fredholm operator]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,15:3 (2013) 100–107 (In Russian)
