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 Mat. Sb. (N.S.), 1971, Volume 86(128), Number 2(10), Pages 299–313 (Mi msb3295)

Analogs of Weyl inequalities and the trace theorem in Banach space

A. S. Markus, V. I. Matsaev

Abstract: Let $A$ be a completely continuous operator acting on the Banach space $\mathfrak B$, let $\{\lambda_j(A)\}$ be the complete system of its eigenvalues (with regard for multiplicity) and let $s_{n+1}(A)$ be the distance from $A$ to the set of all operators of range dimension not greater than $n$. If
$$\sum_{n=1}^\infty s_n(A)\ln(s_n^{-1}(A)+1)<\infty,$$
then $\operatorname{sp}A=\sum\lambda_j(A)$, where $\operatorname{sp}A$ is a functional which is linear on the set of operators satisfying condition (1) (and continuous in a certain topology) and which coincides with its trace for finite-dimensional $A$. The proof of this theorem is based on certain analogs of the famous Weyl inequalities.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 15:2, 299–312

Bibliographic databases:

UDC: 513.881+517.43
MSC: Primary 47B10; Secondary 46H10

Citation: A. S. Markus, V. I. Matsaev, “Analogs of Weyl inequalities and the trace theorem in Banach space”, Mat. Sb. (N.S.), 86(128):2(10) (1971), 299–313; Math. USSR-Sb., 15:2 (1971), 299–312

Citation in format AMSBIB
\Bibitem{MarMat71} \by A.~S.~Markus, V.~I.~Matsaev \paper Analogs of Weyl inequalities and the trace theorem in Banach space \jour Mat. Sb. (N.S.) \yr 1971 \vol 86(128) \issue 2(10) \pages 299--313 \mathnet{http://mi.mathnet.ru/msb3295} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=298460} \zmath{https://zbmath.org/?q=an:0252.47022} \transl \jour Math. USSR-Sb. \yr 1971 \vol 15 \issue 2 \pages 299--312 \crossref{https://doi.org/10.1070/SM1971v015n02ABEH001546} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. G. L. Litvinov, “Traces of linear operators in locally convex spaces”, Funct. Anal. Appl., 13:1 (1979), 60–62
2. W.B. Johnson, H. König, B. Maurey, J.R. Retherford, “Eigenvalues of p-summing and lp-type operators in Banach spaces”, Journal of Functional Analysis, 32:3 (1979), 353
3. Albrecht Pietsch, “Operator Ideals with a Trace”, Math Nachr, 100:1 (1981), 61
4. M. S. Agranovich, “Spectral Boundary Value Problems in Lipschitz Domains for Strongly Elliptic Systems in Banach Spaces $H_p^\sigma$ and $B_p^\sigma$”, Funct. Anal. Appl., 42:4 (2008), 249–267
5. M. Demuth, F. Hanauska, M. Hansmann, G. Katriel, “Estimating the number of eigenvalues of linear operators on Banach spaces”, Journal of Functional Analysis, 2014
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