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

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
$$|\partial^\beta_x\partial^\alpha_\xi a(x,\xi)|\leqslant C_{\alpha\beta}\lambda(x,\xi)\qquad\forall \alpha,\beta,C_{\alpha\beta}>0,$$
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

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
2. Bong T., “Fredholm Property of Linear Difference-Functional Operators”, Dokl. Akad. Nauk, 324:4 (1992), 757–759
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
4. V. G. Kurbatov, “A Remark on Limit Operators”, Funct. Anal. Appl., 30:1 (1996), 56–58
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
6. Rabinovich V., “An Abstract Scheme of the Limit Operator Method and its Applications”, Dokl. Math., 64:3 (2001), 333–336
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
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
9. Yoram Last, Barry Simon, “The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators”, J Anal Math, 98:1 (2006), 183
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+
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
12. Alberto Parmeggiani, “Non-Commutative Harmonic Oscillators and Related Problems”, Milan J. Math, 2014
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