Abstract
For more than 50 years, the author has asked himself why Lorentz spaces are only defined for two parameters. Has this choice been made just for simplicity or is it a natural bound that cannot be exceeded? This question is principal and has nothing to do with usefulness. Now, I discovered a way to produce Lorentz sequence spaces for any finite number of parameters. Having found the right approach, everything turns out to be elementary; the presentation becomes an orgy of mathematical induction. Unfortunately, the new spaces are only of theoretical interest, since we do not know any handy description of their members. This dilemma is, most likely, the reason for the restriction to two, regretted above. However, by the axiom of choice, mathematicians are used to deals with objects that exist only formally; see Banach limits. Therefore, our situation is much more comfortable. It is recommended that, as a first step, readers should have a short glance at the last section, where historical aspects and the interplay between basic concepts are described. Apart from proved theorems, the paper contains many open problems. It is motivated by the same spirit as my very last bibitem in the references. Senior mathematicians should show the way into the future.
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Acknowledgements
Thomas Kühn, my scientific grandson, deserves invaluable gratitude for checking all proofs. His well-considered comments significantly improved the quality of this paper. Among others, he proposed the references [1, 5]. Many thanks are also owed to Dirk Werner for reading large parts of the manuscript and his helpful hints. Once again I thank Fedor Sukochev and Jinghao Huang for the permission to reproduce the proofs of Lemma 10.2 and Theorem 10.3.
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Communicated by M. S. Moslehian.
In friendship, dedicated to the excellent mathematician Fedor Sukochev.
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Pietsch, A. Lorentz spaces depending on more than two parameters. Ann. Funct. Anal. 15, 16 (2024). https://doi.org/10.1007/s43034-023-00313-w
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DOI: https://doi.org/10.1007/s43034-023-00313-w
Keywords
- Lorentz sequence space
- Symmetric sequence ideal
- Shift-monotone sequence ideal
- Operator ideal
- Approximation space
- Real interpolation