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Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces. (English) Zbl 0564.47009
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.

##### MSC:
 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47L30 Abstract operator algebras on Hilbert spaces 47B38 Linear operators on function spaces (general)
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