no. 3
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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     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},
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     volume = {35},
     number = {3},
     year = {1985},
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     mrnumber = {86m:47022},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1020/

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