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Mat. Sb. (N.S.), 1986, Volume 129(171), Number 2, Pages 175–185 (Mi msb1814)  

This article is cited in 12 scientific papers (total in 12 papers)

Pseudodifferential operators on $\mathbf R^n$ and limit operators

B. V. Lange, V. S. Rabinovich


Abstract: The Fredholm property and spectral properties are considered for pseudodifferential operators on $\mathbf R^n$ with symbol satisfying the estimates
\begin{equation} |\partial^\beta_x\partial^\alpha_\xi a(x,\xi)|\leqslant C_{\alpha\beta}\lambda(x,\xi)\qquad\forall \alpha,\beta,C_{\alpha\beta}>0, \end{equation}
where $\lambda(x,\xi)$ is a basic weight function.
As follows from (1), differentiation of the symbol does not improve its behavior at infinity.
The family of limit operators is introduced for a pseudodifferential operator. A theorem is proved giving necessary and sufficient conditions for the Fredholm property in terms of invertibility of the family of limit operators. Some properties of the spectrum are formulated in the same terms. Examples are given which illustrate the main results.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 57:1, 183–194

Bibliographic databases:

UDC: 517.9
MSC: Primary 35S05, 47A53; Secondary 47G05
Received: 30.01.1984

Citation: B. V. Lange, V. S. Rabinovich, “Pseudodifferential operators on $\mathbf R^n$ and limit operators”, Mat. Sb. (N.S.), 129(171):2 (1986), 175–185; Math. USSR-Sb., 57:1 (1987), 183–194

Citation in format AMSBIB
\Bibitem{LanRab86}
\by B.~V.~Lange, V.~S.~Rabinovich
\paper Pseudodifferential operators on $\mathbf R^n$ and limit operators
\jour Mat. Sb. (N.S.)
\yr 1986
\vol 129(171)
\issue 2
\pages 175--185
\mathnet{http://mi.mathnet.ru/msb1814}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=832115}
\zmath{https://zbmath.org/?q=an:0658.47051|0611.47039}
\transl
\jour Math. USSR-Sb.
\yr 1987
\vol 57
\issue 1
\pages 183--194
\crossref{https://doi.org/10.1070/SM1987v057n01ABEH003063}


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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. Rabinovich V., “Fredholmness of Boundary-Value-Problems on Noncompact Manifolds and Limiting Operators”, Dokl. Akad. Nauk, 325:2 (1992), 237–241  mathnet  mathscinet  zmath  isi
    2. Bong T., “Fredholm Property of Linear Difference-Functional Operators”, Dokl. Akad. Nauk, 324:4 (1992), 757–759  mathnet  mathscinet  zmath  isi
    3. Rabinovich V., “Criterion of a Local Invertibility of the Mellin Pseudodifferential-Operators and Some its Applications”, Dokl. Akad. Nauk, 333:2 (1993), 147–150  mathnet  mathscinet  zmath  isi
    4. V. G. Kurbatov, “A Remark on Limit Operators”, Funct. Anal. Appl., 30:1 (1996), 56–58  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Rabinovich V., Roch S., Silbermann B., “Fredholm Theory and Finite Section Method for Band-Dominated Operators”, Integr. Equ. Oper. Theory, 30:4 (1998), 452–495  crossref  mathscinet  zmath  isi
    6. Rabinovich V., “An Abstract Scheme of the Limit Operator Method and its Applications”, Dokl. Math., 64:3 (2001), 333–336  mathscinet  zmath  isi
    7. Karapetyants A., Rabinovich V., Vasilevski N., “On Algebras of Two Dimensional Singular Integral Operators with Homogeneous Discontinuities in Symbols”, Integr. Equ. Oper. Theory, 40:3 (2001), 278–308  crossref  mathscinet  zmath  isi
    8. Rabinovich V., Roch S., “Integral Operators with Shifts on Homogeneous Groups”, Factorization, Singular Operators and Related Problems, Proceedings, eds. Samko S., Lebre A., DosSantos A., Springer, 2003, 255–271  crossref  mathscinet  isi
    9. Yoram Last, Barry Simon, “The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators”, J Anal Math, 98:1 (2006), 183  crossref  mathscinet  zmath  isi  elib
    10. Chandler-Wilde S.N. Lindner M., “Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices”, Mem. Am. Math. Soc., 210:989 (2011), 1+  crossref  mathscinet  isi  elib
    11. Yuri I. Karlovich, Iván Loreto Hernández, “Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study”, Integr. Equ. Oper. Theory, 2012  crossref  mathscinet
    12. Alberto Parmeggiani, “Non-Commutative Harmonic Oscillators and Related Problems”, Milan J. Math, 2014  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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