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Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces

Bernd Carl — 1985

Annales de l'institut Fourier

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...

Local entropy moduli and eigenvalues of operators in Banach spaces.

Bernd CarlThomas Kühn — 1985

Revista Matemática Iberoamericana

In the paper local entropy moduli of operators between Banach spaces are introduced. They constitue a generalization of entropy numbers and moduli, and localize these notions in an appropriate way. Many results regarding entropy numbers and moduli can be carried over to local entropy moduli. We investigate relations between local entropy moduli and s-numbers, spectral properties, eigenvalues, absolutely summing operators. As applications, local entropy moduli of identical and diagonal operators...

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd CarlDavid E. Edmunds — 2003

Studia Mathematica

For a precompact subset K of a Hilbert space we prove the following inequalities: n 1 / 2 c ( c o v ( K ) ) c K ( 1 + k = 1 k - 1 / 2 e k ( K ) ) , n ∈ ℕ, and k 1 / 2 c k + n ( c o v ( K ) ) c [ l o g 1 / 2 ( n + 1 ) ε ( K ) + j = n + 1 ε j ( K ) / ( j l o g 1 / 2 ( j + 1 ) ) ] , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and ε k ( K ) and e k ( K ) denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly...

Weyl numbers versus Z-Weyl numbers

Bernd CarlAndreas DefantDoris Planer — 2014

Studia Mathematica

Given an infinite-dimensional Banach space Z (substituting the Hilbert space ℓ₂), the s-number sequence of Z-Weyl numbers is generated by the approximation numbers according to the pattern of the classical Weyl numbers. We compare Weyl numbers with Z-Weyl numbers-a problem originally posed by A. Pietsch. We recover a result of Hinrichs and the first author showing that the Weyl numbers are in a sense minimal. This emphasizes the outstanding role of Weyl numbers within the theory of eigenvalue distribution...

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