scholarly journals Some new bounds on the entropy numbers of diagonal operators

2020 ◽  
Vol 251 ◽  
pp. 105343
Author(s):  
Simon Fischer
Keyword(s):  
Entropy Numbers ◽  
Diagonal Operators

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

ENTROPY NUMBERS OF DIAGONAL OPERATORS BETWEEN VECTOR-VALUED SEQUENCE SPACES

2001 ◽  
Vol 64 (3) ◽  
pp. 739-754 ◽  
Author(s):  
THOMAS KÜHN ◽  
TOMAS P. SCHONBEK
Keyword(s):  
Lower Bounds ◽  
Sequence Spaces ◽  
Upper And Lower Bounds ◽  
Entropy Numbers ◽  
Diagonal Operators ◽  
Weighted Sequence ◽  
Banach Sequence Spaces ◽  
Vector Valued ◽  
Banach Sequence

Upper and lower bounds are established for the entropy numbers of certain diagonal operators between Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question coincide with those considered by Leopold. The results in the paper improve the previous results in at least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the upper estimates are improvements on the previous ones, the improvement being quite significant in some cases.

scholarly journals Entropy Numbers of Diagonal Operators of Logarithmic Type

2001 ◽  
Vol 8 (2) ◽  
pp. 307-318
Author(s):  
Thomas Kühn
Keyword(s):  
Asymptotic Behaviour ◽  
Entropy Numbers ◽  
Diagonal Operators ◽  
Logarithmic Type

Abstract We determine the asymptotic behaviour (as 𝑘 → ∞, up to multiplicative constants not depending on k) of the entropy numbers 𝑒𝑘 (D σ : l p → l q ), 1 ≤ p ≤ q ≤ ∞, of diagonal operators generated by logarithmically decreasing sequences σ = (σ n ). This complements earlier results by Carl [J. Approx. Theory 32: 135–150, 1981] who investigated the case of power-like decay of the diagonal.

Sign in / Sign up

Export Citation Format

Share Document