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Entropy and n-widths of operators in Banach spaces

100%

Carl B.

Banach Center Publications

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1989

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tom 22

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nr 1

55-67

2

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

84%

Carl B., Edmunds D.

Studia Mathematica

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2003

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tom 159

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nr 3

391-402

EN

For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1/2} cₙ(cov(K)) ≤ c_{K}(1 + ∑^{ⁿ}_{k=1} k^{-1/2} e_k(K))$, n ∈ ℕ, and $k^{1/2} c_{k+n}(cov(K)) ≤ c[log^{1/2}(n+1)εₙ(K) + ∑_{j=n+1}^{∞} ε_j(K)/(j log^{1/2}(j+1))]$, 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 decreasing. For example, we get optimal estimates in the non-critical case where $εₙ(K) ⪯ log^{-α}(n + 1)$, α ≠ 1/2, 0 < α < ∞, as well as in the critical case where α = 1/2. For α = 1/2 we show the asymptotically optimal estimate $cₙ(cov(K)) ⪯ n^{-1/2} log(n + 1)$, which refines the corresponding result of Gao [Ga] obtained for entropy numbers. Furthermore, we establish inequalities similar to that of Creutzig and Steinwart [CrSt] in the critical as well as non-critical cases. Finally, we give an alternative proof of a result by Li and Linde [LL] for Gelfand and entropy numbers of the absolutely convex hull of K when K has the shape K = {t₁,t₂,...}, where ||tₙ|| ≤ σₙ, σₙ↓ 0. In particular, for $σₙ ≤ log^{-1/2}(n + 1)$, which corresponds to the critical case, we get a better asymptotic behaviour of Gelfand numbers, $cₙ(cov(K)) ⪯ n^{-1/2}$.

3

Weyl numbers versus Z-Weyl numbers

68%

Carl B., Defant A., Planer D.

Studia Mathematica

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2014

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tom 223

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nr 3

233-250

EN

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 of operators between Banach spaces.

4

Entropy numbers of r-nuclear operators between $L_{p}$ spaces

42%

Carl B.

Studia Mathematica

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1983-1984

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tom 77

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nr 2

155-162

Ograniczanie wyników

3 Studia Mathematica

1 Banach Center Publications

4 Carl B.

1 Defant A.

1 Edmunds D.

1 Planer D.

1 2014

1 2003

1 1989

1 1983

Entropy and n-widths of operators in Banach spaces

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Weyl numbers versus Z-Weyl numbers

Entropy numbers of r-nuclear operators between $L_{p}$ spaces