Uniqueness of unconditional basis of $\ell _{2}\oplus \mathcal {T}^{(2)}$

F. Albiac, J. L. Ansorena
{"title":"Uniqueness of unconditional basis of $\\ell _{2}\\oplus \\mathcal {T}^{(2)}$","authors":"F. Albiac, J. L. Ansorena","doi":"10.1090/PROC/15670","DOIUrl":null,"url":null,"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 its 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.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/PROC/15670","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

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 its 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.
$\ well _{2}\oplus \mathcal {T}^{(2)}$的无条件基的唯一性
我们提供了准Banach格间紧算子的Pitt定理的一个新推广,它允许将Banach空间$\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$的有限直接和的无条件基描述为其和的无条件基的直接和。我们得到一般的分裂原理,特别是,如果每个$\mathbb{X}_{i}$有一个唯一的无条件基(直到等价和置换),那么$\mathbb{X}_{1}\oplus \cdots\oplus\mathbb{X}_{n}$也有一个唯一的无条件基。在我们的技术对巴拿赫和拟巴拿赫空间结构的新应用中,我们发现空间$\ell_2\oplus \mathcal{T}^{(2)}$具有唯一的无条件基。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信