Uniqueness of unconditional basis of $\ell _{2}\oplus \mathcal T^{(2)}$
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- by Fernando Albiac and José L. Ansorena PDF
- Proc. Amer. Math. Soc. 150 (2022), 709-717 Request permission
Abstract:
We provide a new extension of Pitt’s theorem for compact operators between quasi-Banach lattices which permits to describe unconditional bases of finite direct sums of Banach spaces $\mathbb {X}_{1}\oplus \dots \oplus \mathbb {X}_{n}$ as direct sums of unconditional bases of their summands. The general splitting principle we obtain yields, in particular, that if each $\mathbb {X}_{i}$ has a unique unconditional basis (up to equivalence and permutation), then $\mathbb {X}_{1}\oplus \cdots \oplus \mathbb {X}_{n}$ has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $\ell _2\oplus \mathcal {T}^{(2)}$ has a unique unconditional basis.References
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Additional Information
- Fernando 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
- José L. Ansorena
- Affiliation: Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño 26004, Spain
- MR Author ID: 359480
- ORCID: 0000-0002-4979-1080
- Email: joseluis.ansorena@unirioja.es
- Received by editor(s): October 19, 2020
- Received by editor(s) in revised form: May 13, 2021
- Published electronically: November 15, 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. Both authors acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación
- Communicated by: Stephen Dilworth
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 709-717
- MSC (2020): Primary 46B15, 46B20, 46B42, 46B45, 46A16, 46A35, 46A40, 46A45
- DOI: https://doi.org/10.1090/proc/15670
- MathSciNet review: 4356181