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

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.

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.

@article{AIF_1985__35_3_79_0,
     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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {35},
     number = {3},
     year = {1985},
     pages = {79-118},
     doi = {10.5802/aif.1020},
     zbl = {0564.47009},
     mrnumber = {86m:47022},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1985__35_3_79_0}
}
Carl, Bernd. Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces. Annales de l'Institut Fourier, Volume 35 (1985) no. 3, pp. 79-118. doi : 10.5802/aif.1020. http://www.numdam.org/item/AIF_1985__35_3_79_0/

[1] B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal., 41 (1981), 290-306. | MR 82m:47015 | Zbl 0466.41008

[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 84i:47033 | Zbl 0509.47017

[3] B. Carl, Entropy numbers of r-nuclear operators between Lp spaces, Studia Math., 77 (1983), 155-162. | MR 85e:47028 | Zbl 0563.47013

[4] B. Carl, On the degree of compactness of operators acting from function spaces into Banach spaces of type q, (Jena 1982). | Zbl 0507.47015

[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 82b:47022 | Zbl 0465.47019

[6] X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, Lecture Notes Math., 480 (1975), 1-96. | MR 54 #1355 | Zbl 0331.60025

[7] T. Figiel, J. Lindenstrauss, V.D. Milman, The dimensions of almost spherical sections of convex bodies, Acta Math., 139 (1977), 53-94. | MR 56 #3618 | Zbl 0375.52002

[8] E.D. Gluskin, On some finite dimensional problems of the theory of diameters, Vestnik Leningr. Univ., 13 (1981), 5-10. | MR 83d:46018 | Zbl 0482.41018

[9] E.D. Gluskin, Norms of random matrices and diameters of finite dimensional sets, Math. Sbornic, 120 (1983), 180-189. | MR 84g:41021 | Zbl 0528.46015

[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 81b:42002 | Zbl 0411.41006

[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 50 #8626 | Zbl 0265.60005

[13] R.A. Hunt, On L (p, q) spaces, Enseign. Math., 12 (1966), 249-276. | MR 36 #6921 | Zbl 0181.40301

[14] W.B. Johnson, G. Schechtman, Embedding lmp into ln1, Acta Math., 149 (1982), 71-85. | MR 84a:46031 | Zbl 0522.46015

[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 0574.47018

[17] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Lect. Notes Math., 338, Berlin - Heidelberg - New York, 1973. | MR 54 #3344 | Zbl 0259.46011

[18] G.G. Lorentz, Approximation of Functions, Academic Press, New York/Toronto/London, 1966. | MR 35 #4642 | Zbl 0153.38901

[19] E. Makai Jr., J. Zemanek, Geometrical means of eigenvalues, J. Operator Theory, 7 (1982), 173-178. | MR 83m:47005 | Zbl 0483.47018

[20] M. Marcus, G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes (to appear). | Zbl 0547.60047

[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 56 #1388 | Zbl 0344.47014

[22] B.S. Mitjagin, A. Pełczynski, Nuclear operators and approximative dimension, Proc. ICM, (1966), 366-372. | Zbl 0191.41704

[23] A. Pietsch, Operator ideals, Berlin, 1978. | MR 81a:47002 | Zbl 0399.47039

[24] A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann., 47 (1980), 149-168. | MR 82i:47073a | Zbl 0428.47027

[25] G. Pisier, Remarques sur un résultat non public de B. Maurey, Sem. d'Analyse Fonctionnelle 1980/1981, Exp. V. | Numdam | Zbl 0491.46017

[26] G. Pisier, On the dimension of the lnp-subspaces of Banach spaces, for 1 ≤ p < 2, Trans. AMS, 276 (1983), 201-211. | MR 84a:46035 | Zbl 0509.46016

[27] F. Riesz, Über lineare Funktionalgleichungen, Acta Math., 41 (1918), 71-98. | JFM 46.0635.01

[28] J. Schauder, Über lineare, vollstetige Funktionaloperationen, Studia Math., 2 (1930), 1-6. | JFM 56.0353.02

[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 0395.46015

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

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