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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

Full text: PDF file (549 kB)
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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

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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  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    4. M. S. Bichegkuev, “Spectral analysis of difference and differential operators in weighted spaces”, Sb. Math., 204:11 (2013), 1549–1564  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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