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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

AnnotationIn the monograph \cite{kyashkinb1} and the article \cite{kyashkinb2} the problem on perturbation of linear equation by small linear summand of the form $(B-\varepsilon A)x=h$ were investigated with closely defined on $D_{B}$ Fredholmian operator $B:E_{1}\supset D_{B}\rightarrow E_{2}$, $D_{A}\supset D_{B}$, or $A\in L\{E_{1},E_{2}\}$, $\varepsilon\in\mathbb{C}^{1}$ - small parameter, $E_{1}$ and $E_{2}$ - 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 $\varepsilon$ operator-function $A(\varepsilon)$. 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 $A(0)$.
Keywordslinear 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)