Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces
Annales de l'Institut Fourier, Tome 35 (1985) no. 3, pp. 79-118.

Dans cet article sont traités des problèmes de recouvrement et le degré de compacité des opérateurs. La part essentielle est consacrée aux relations entre les modules d’entropie et les nombres de Kolmogoroff (ou plutôt de Gelfand et d’approximation) des opérateurs qui peuvent être interprétés comme un pendant pour les inégalités classiques de Bernstein-Jackson pour les fonctions. Quelques quantifications des résultats de la théorie de Riesz-Schauder sont données. Enfin, la plus grande distance entre le “degré d’approximation” et le “degré de compacité” des opérateurs intégraux en C[0,1] engendrés par des noyaux lisses est déterminée. Pour illustrer les quantifications nous traitons quelques problèmes de valeurs propres et de compacité des opérateurs nucléaires et du type Hille-Tamarkin.

The paper deals with covering problems and the degree of compactness of operators. The main part is devoted to relationships between entropy moduli and Kolmogorov (resp. Gelfand and approximation) numbers for operators which may be interpreted as counterparts to the classical Bernstein-Jackson inequalities for functions. Certain quantifications of results in the Riesz-Schauder-Theory are given. Finally, the largest distance between “the degree of approximation” and the “degree of compactness” of integral operators in C[0,1] generated by smooth kernels is determined. For illustrating of the quantifications we treat some eigenvalue and compactness problems of nuclear operators and operators of Hille-Tamarkin-type.

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     author = {Carl, Bernd},
     title = {Inequalities of {Bernstein-Jackson-type} and the degree of compactness of operators in {Banach} spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {79--118},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {35},
     number = {3},
     year = {1985},
     doi = {10.5802/aif.1020},
     mrnumber = {86m:47022},
     zbl = {0564.47009},
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     url = {http://www.numdam.org/articles/10.5802/aif.1020/}
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Carl, Bernd. Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces. Annales de l'Institut Fourier, Tome 35 (1985) no. 3, pp. 79-118. doi : 10.5802/aif.1020. http://www.numdam.org/articles/10.5802/aif.1020/

[1] B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal., 41 (1981), 290-306. | MR | Zbl

[2] B. Carl, On a characterization of operators from lq into a Banach space of type p with some applications to eigenvalue problems, J. Funct. Anal., 48 (1982), 394-407. | MR | Zbl

[3] B. Carl, Entropy numbers of r-nuclear operators between Lp spaces, Studia Math., 77 (1983), 155-162. | MR | Zbl

[4] B. Carl, On the degree of compactness of operators acting from function spaces into Banach spaces of type q, (Jena 1982). | Zbl

[5] B. Carl, H. Triebel, Inequalities between eigenvalues, entropy numbers and related quantities of compact operators in Banach spaces, Math. Ann., 251 (1980), 129-133. | MR | Zbl

[6] X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, Lecture Notes Math., 480 (1975), 1-96. | MR | Zbl

[7] T. Figiel, J. Lindenstrauss, V.D. Milman, The dimensions of almost spherical sections of convex bodies, Acta Math., 139 (1977), 53-94. | MR | Zbl

[8] E.D. Gluskin, On some finite dimensional problems of the theory of diameters, Vestnik Leningr. Univ., 13 (1981), 5-10. | MR | Zbl

[9] E.D. Gluskin, Norms of random matrices and diameters of finite dimensional sets, Math. Sbornic, 120 (1983), 180-189. | MR | Zbl

[10] U. Haagerup, The best constants in the Khintchine inequality, Proc. Intern. Conf. “Operator algebras, ideals, ...”, Teubner Texte Math., pp. 69-79, Leipzig, 1978. | MR | Zbl

[11] S. Heinrich, Optimal approximation of integral operators, in preparation.

[12] J. Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math., 52 (1974), 159-186. | MR | Zbl

[13] R.A. Hunt, On L (p, q) spaces, Enseign. Math., 12 (1966), 249-276. | MR | Zbl

[14] W.B. Johnson, G. Schechtman, Embedding lmp into ln1, Acta Math., 149 (1982), 71-85. | MR | Zbl

[15] B.S. Kashin, Sections of some finite dimensional sets and classes of smooth functions, Izv. ANSSR, ser. mat., 41 (1977), 334-351, (Russian).

[16] T. Kühn, Entropy numbers of r-nuclear operators in Banach spaces of type, Studia Math., (to appear). | Zbl

[17] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Lect. Notes Math., 338, Berlin - Heidelberg - New York, 1973. | MR | Zbl

[18] G.G. Lorentz, Approximation of Functions, Academic Press, New York/Toronto/London, 1966. | MR | Zbl

[19] E. Makai Jr., J. Zemanek, Geometrical means of eigenvalues, J. Operator Theory, 7 (1982), 173-178. | MR | Zbl

[20] M. Marcus, G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes (to appear). | Zbl

[21] B. Maurey, G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math., 58 (1976), 45-90. | MR | Zbl

[22] B.S. Mitjagin, A. Pełczynski, Nuclear operators and approximative dimension, Proc. ICM, (1966), 366-372. | Zbl

[23] A. Pietsch, Operator ideals, Berlin, 1978. | MR | Zbl

[24] A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann., 47 (1980), 149-168. | MR | Zbl

[25] G. Pisier, Remarques sur un résultat non public de B. Maurey, Sem. d'Analyse Fonctionnelle 1980/1981, Exp. V. | Numdam | Zbl

[26] G. Pisier, On the dimension of the lnp-subspaces of Banach spaces, for 1 ≤ p < 2, Trans. AMS, 276 (1983), 201-211. | MR | Zbl

[27] F. Riesz, Über lineare Funktionalgleichungen, Acta Math., 41 (1918), 71-98. | JFM

[28] J. Schauder, Über lineare, vollstetige Funktionaloperationen, Studia Math., 2 (1930), 1-6. | JFM

[29] C. Schutt, Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory (to appear).

[30] J.S. Szarck, On Kashin's almost euclidean orthogonal decomposition of l1n, Bull. Acad. Polon. Sci., 26 (1978). | Zbl

[31] A.F. Timan, Approximation Theory of functions of Real Variables, Moscow, 1960.

[32] A. Zygmund, Trigonometric Series, Cambridge, 1968.

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