scholarly journals Entropy Numbers of Vector-Valued Diagonal Operators

2002 ◽  
Vol 117 (1) ◽  
pp. 132-139 ◽  
Author(s):  
Eduard Belinsky
Keyword(s):  
Entropy Numbers ◽  
Diagonal Operators ◽  
Vector Valued

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.

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 operators acting between vector-valued sequence spaces

2012 ◽  
Vol 286 (5-6) ◽  
pp. 614-630 ◽  
Author(s):  
David E. Edmunds ◽  
Yuri Netrusov
Keyword(s):  
Sequence Spaces ◽  
Entropy Numbers ◽  
Vector Valued

scholarly journals An operator ideal generated by Orlicz spaces

2019 ◽  
Vol 376 (3-4) ◽  
pp. 1675-1703
Author(s):  
Mieczysław Mastyło
Keyword(s):  
Hilbert Space ◽  
Wide Class ◽  
Orlicz Spaces ◽  
Asymptotic Estimates ◽  
Approximation Numbers ◽  
Diagonal Operators ◽  
Finite Dimensional ◽  
Rademacher Series ◽  
Vector Valued ◽  
Banach Sequence

Abstract Absolutely $$\varphi $$φ-summing operators between Banach spaces generated by Orlicz spaces are investigated. A variant of Pietsch’s domination theorem is proved for these operators and applied to prove vector-valued inequalities. These results are used to prove asymptotic estimates of $$\pi _\varphi $$πφ-summing norms of finite-dimensional operators and also diagonal operators between Banach sequence lattices for a wide class of Orlicz spaces based on exponential convex functions $$\varphi $$φ. The key here is the description of a space of coefficients of the Rademacher series in this class of Orlicz spaces, proved via interpolation methods. As by-product, some absolutely $$\varphi $$φ-summing operators on the Hilbert space $$\ell _2$$ℓ2 are characterized in terms of its approximation numbers.

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