Geodesic
Analogs of Weyl inequalities and the trace theorem in Banach space
Sbornik. Mathematics, Tome 15 (1971) no. 2, pp. 299-312 
A. S. Markus; V. I. Matsaev. Analogs of Weyl inequalities and the trace theorem in Banach space. Sbornik. Mathematics, Tome 15 (1971) no. 2, pp. 299-312. http://geodesic.mathdoc.fr/item/SM_1971_15_2_a6/
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     title = {Analogs of {Weyl} inequalities and the trace theorem in {Banach} space},
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     volume = {15},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_15_2_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 \begin{equation} \sum_{n=1}^\infty s_n(A)\ln\bigl(s_n^{-1}(A)+1\bigr)<\infty, \end{equation} 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.

[1] R. Schatten, A theory of cross-spaces, Princeton, 1950 | MR

[2] A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc., no. 16, Providence, 1955 | MR

[3] P. Pietsch, “Einige neue Klassen von kompakten linearen Abbildungen”, Rev. Math. Pures et Appl., 8:3 (1963), 427–447 | MR | Zbl

[4] V. B. Lidskii, “Nesamosopryazhennye operatory, imeyuschie sled”, DAN SSSR, 125:3 (1959), 485–487 | MR

[5] H. Weyl, “Inequalities between the two kinds of eigenvalues of a linear transformation”, Proc. Nat. Acad. Sci. USA, 35:7 (1949), 408–411 | DOI | MR | Zbl

[6] A. Pich, Yadernye lokalno vypuklye prostranstva, Mir, Moskva, 1967 | MR

[7] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, Moskva, 1965

[8] F. John, “Extremum problems with inequalities as subsidiary conditions”, Courant Anniversary Volume, Interscience, New York, 1948, 186–204 | MR

[9] M. I. Kadets, M. G. Snobar, “O nekotorykh funktsionalakh na kompakte Minkovskogo”, Matem. zametki, 10:4 (1971) | MR | Zbl

[10] A. S. Markus, V. I. Paraska, “Ob otsenke chisla sobstvennykh znachenii lineinogo operatora”, Izv. AN Mold. SSR, 1965, no. 7, 101–104 | MR | Zbl

[11] E. I. Sigal, “O kratnosti kharakteristicheskogo chisla proizvedeniya operator-funktsii”, Matem. issl., 5:1 (1970), 118–127 | MR | Zbl

[12] A. S. Markus, “Nekotorye priznaki polnoty sistemy kornevykh vektorov lineinogo operatora v banakhovom prostranstve”, Matem. sb., 70(112) (1966), 526–561 | MR

[13] S. Kwapień, A. Pelczyński, “The main triangle projection in matrix spaces and its applications”, Studia Math., 34:1 (1970), 43–67 | MR

[14] E. R. Lorch, “Bicontinuous linear transformations in certain vector spaces”, Bull. Amer. Math. Soc., 45:8 (1939), 564–569 | DOI | MR | Zbl