Abstract
We prove the following dichotomy for the spaces ℒ (a)p,q,α (X, Y) of all operators T ∈ ℒ(X, Y) whose approximation numbers belong to the Lorentz-Zygmund sequence spaces ℓp,q(log ℓ)α: If X and Y are infinite-dimensional Banach spaces, then the spaces ℒ (a)p,q,α (X, Y) with 0 < p < ∞, 0 < q ≤ ∞ and α ∈ ℝ are all different from each other, but otherwise, if X or Y are finite-dimensional, they are all equal (to ℒ(X, Y)).
Moreover we show that the scale \({\{ {\cal L}_{\infty ,q}^{(a)}(X,Y)\} _{0\, < q\, < \infty }}\) is strictly increasing in q, where ℒ (a)∈,q (X, Y) is the space of all operators in ℒ(X, Y) whose approximation numbers are in the limiting Lorentz sequence space ∓∈,q.
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Acknowledgements
The authors would like to thank Prof. Albrecht Pietsch for some stimulating mails and Leo R. Ya. Doktorski for pointing out to us the paper [1].
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Dedicated to Professor Oleg Vladimirovich Besov on the occasion of his 90th birthday
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Cobos, F., Kühn, T. Diversity of Lorentz-Zygmund Spaces of Operators Defined by Approximation Numbers. Anal Math 49, 951–969 (2023). https://doi.org/10.1007/s10476-023-0239-x
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DOI: https://doi.org/10.1007/s10476-023-0239-x
Key words and phrases
- space of operators defined by approximation numbers
- dependence on the parameters
- logarithmic interpolation space
- Lorentz-Zygmund space