A simple embedding theorem for kernels of trace class integral operators in $L^2(\mathbb {R}^m)$. Application to the Fredholm trace formula
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M. Sh. Birman
Translated by: Ari Laptev - St. Petersburg Math. J. 27 (2016), 327-331
- DOI: https://doi.org/10.1090/spmj/1389
- Published electronically: January 29, 2016
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Abstract:
The present paper by Mikhail Shlemovich Birman was written in 1989 and circulated among specialists as a preprint published in English by Linköping University (the original manuscript of M. Sh. Birman was translated by A. A. Laptev). In this paper a transparent approach to the proof of the Fredholm formula for the traces of integral operators of trace class was found. By communication with D. R. Yafaev we knew that M. Sh. Birman did not publish this paper because he discovered that a similar construction was used in the book by M. A. Shubin on pseudodifferential operators. This is so, but the presentation in the present text is much more general, clear, and detailed. In this connection, and also in connection with the renewed interest to integral formulas for traces of integral operators, the editorial board decided to publish this paper under the heading “Easy Reading for Professionals”.References
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- M. Sh. Birman and M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert operators, 2nd ed., revised and augmented, Lan′, St. Petersbug, 2010. (Russian)
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- M. Sh. Birman and D. R. Yafaev, A general scheme in the stationary theory of scattering, Wave propagation. Scattering theory (Russian), Probl. Mat. Fiz., vol. 12, Leningrad. Univ., Leningrad, 1987, pp. 89–117, 257–258 (Russian). MR 923973
- M. Š. Birman and M. Z. Solomjak, Estimates for the singular numbers of integral operators, Uspehi Mat. Nauk 32 (1977), no. 1(193), 17–84, 271 (Russian). MR 0438186
Bibliographic Information
- Received by editor(s): April 1, 2014
- Published electronically: January 29, 2016
- Additional Notes: During the lifetime of the author this text was published as the preprint: Birman M. Sh., A proof of the Fredholm trace formula as an application of a simple embedding for kernels of integral operators of trace class in $L^2({\mathbb R}^m)$, Preprint, Dept. Math., Inst. Technol., Linköping Univ., LITH-MAT-R-89-30, 1989
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 327-331
- MSC (2010): Primary 47B10
- DOI: https://doi.org/10.1090/spmj/1389
- MathSciNet review: 3444466
Dedicated: Easy reading for professionals