@article{d8cf1b84fa5c4dd0b8d310882543c432,
title = "A dichotomy for subsymmetric basic sequences with applications to garling spaces",
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.",
keywords = "Garling spaces, Lorentz spaces, Sequence spaces, Subsymmetric basic sequence, Symmetric basic sequence",
author = "F. Albiac and Ansorena, {J. L.} and Dilworth, {S. J.} and Denka Kutzarova",
note = "Funding Information: 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. Funding Information: 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 An{\'a}lisis Vectorial, Multilineal y Aproximaci{\'o}n. 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. Funding Information: 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. Funding Information: The second author acknowledges the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for An{\'a}lisis Vectorial, Multilineal y Aproximaci{\'o}n. Publisher Copyright: {\textcopyright} 2021 American Mathematical Society. All rights reserved.",
year = "2021",
month = mar,
doi = "10.1090/tran/8278",
language = "English (US)",
volume = "374",
pages = "2079--2106",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "3",
}