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 Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 4, Pages 3–20 (Mi izv4458)

This article is cited in 8 scientific papers (total in 8 papers)

On conditions for invertibility of difference and differential operators in weight spaces

M. S. Bichegkuev

North-Ossetia State University

Abstract: We obtain necessary and sufficient conditions for the invertibility of the difference operator $\mathcal{D}_E\colon D(\mathcal{D}_E)\subset l^p_\alpha \to l^p_\alpha$, $(\mathcal{D}_E x)(n)=x(n+1)-Bx(n)$, $n\in \mathbb{Z}_+$, whose domain $D(\mathcal{D}_E)$ is given by the condition $x(0)\in E$, where $l^p_\alpha=l^p_\alpha(\mathbb{Z}_+,X)$, $p\in[1,\infty]$, is the Banach space of sequences (of vectors in a Banach space $X$) summable with weight $\alpha\colon\mathbb{Z}_+\to (0,\infty)$ for $p\in[1,\infty)$ and bounded with respect to $\alpha$ for $p=\infty$, $B\colon X\to X$ is a bounded linear operator, and $E$ is a closed $B$-invariant subspace of $X$. We give applications to the invertibility of differential operators with an unbounded operator coefficient (the generator of a strongly continuous operator semigroup) in weight spaces of functions.

Keywords: difference operator, spectrum of an operator, invertible operator, weight spaces of sequences and functions, linear relation, differential operator.

DOI: https://doi.org/10.4213/im4458

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English version:
Izvestiya: Mathematics, 2011, 75:4, 665–680

Bibliographic databases:

UDC: 517.9
MSC: 47B37, 47B39
Received: 11.02.2010
Revised: 18.11.2010

Citation: M. S. Bichegkuev, “On conditions for invertibility of difference and differential operators in weight spaces”, Izv. RAN. Ser. Mat., 75:4 (2011), 3–20; Izv. Math., 75:4 (2011), 665–680

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bichegkuev M.S., “On the exponential dichotomy and spectral properties of difference operators related to the howland semigroup”, Differ. Equ., 48:6 (2012), 769–778
2. A. G. Baskakov, “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations”, Russian Math. Surveys, 68:1 (2013), 69–116
3. Todorov D., “Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory”, Discrete Contin. Dyn. Syst., 33:9 (2013), 4187–4205
4. M. S. Bichegkuev, “Spectral analysis of difference and differential operators in weighted spaces”, Sb. Math., 204:11 (2013), 1549–1564
5. Baskakov A.G. Krishtal I.A., “On Completeness of Spectral Subspaces of Linear Relations and Ordered Pairs of Linear Operators”, J. Math. Anal. Appl., 407:1 (2013), 157–178
6. M. S. Bichegkuev, “Spectral Analysis of Differential Operators with Unbounded Operator Coefficients in Weighted Spaces of Functions”, Math. Notes, 95:1 (2014), 15–21
7. V. M. Bruk, “On Linear Relations Generated by an Integro-Differential Equation with Nevanlinna Measure in the Infinite-Dimensional Case”, Math. Notes, 96:1 (2014), 10–25
8. V. M. Bruk, “Dissipative expansions of a symmetric relation generated by a system of integral equations with operator measures”, Russian Math. (Iz. VUZ), 58:12 (2014), 7–22
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