A dichotomy for subsymmetric basic sequences with applications to Garling spaces
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- by F. Albiac, J. L. Ansorena, S. J. Dilworth and Denka Kutzarova PDF
- Trans. Amer. Math. Soc. 374 (2021), 2079-2106 Request permission
Abstract:
Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positionings and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.References
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Additional Information
- F. Albiac
- Affiliation: Department of Mathematics, Statistics and Computer Sciences, and InaMat2, Universidad Pública de Navarra, Pamplona 31006, Spain
- MR Author ID: 692748
- ORCID: 0000-0001-7051-9279
- Email: fernando.albiac@unavarra.es
- J. L. Ansorena
- Affiliation: Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño 26004, Spain
- MR Author ID: 359480
- Email: joseluis.ansorena@unirioja.es
- S. J. Dilworth
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 58105
- Email: dilworth@math.sc.edu
- Denka Kutzarova
- Affiliation: Department of Mathematics University of Illinois at Urbana-Champaign, Urbana, Illinois 61801; and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
- MR Author ID: 108570
- Email: denka@math.uiuc.edu
- Received by editor(s): October 30, 2019
- Received by editor(s) in revised form: July 10, 2020, and August 1, 2020
- Published electronically: January 12, 2021
- Additional Notes: The first author acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces and the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. He would like to thank the Isaac Newton Institute, Cambridge, for support and hospitality during the programme Approximation, Sampling and Compression in Data Science, where work on this paper was undertaken. This work was supported by ESPRC Grant no EP/K032208/1.
The second author acknowledges the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación.
The third author acknowledges support from the National Science Foundation under Grant Number DMS–1361461. He would like to thank the Isaac Newton Institute, Cambridge, for support and hospitality during the programme Approximation, Sampling and Compression in Data Science, where work on this paper was undertaken. This work was supported by ESPRC Grant no EP/K032208/1.
The fourth author acknowledges the support from the Simons Foundation Collaborative under Grant No 636954. She would like to thank the Isaac Newton Institute, Cambridge, for support and hospitality during the programme Approximation, Sampling and Compression in Data Science, where work on this paper was undertaken. This work was supported by ESPRC Grant no EP/K032208/1. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 2079-2106
- MSC (2020): Primary 46B15; Secondary 46B20, 46B45
- DOI: https://doi.org/10.1090/tran/8278
- MathSciNet review: 4216733