On the structure of the set of higher order spreading models
type: Item
instance of: scholarly article
title: On the structure of the set of higher order spreading models
zbMATH Open document ID: 1322.46012
author: Bünyamin Sarı
author: Konstantinos Tyros
published in: Studia Mathematica
publication date: 2014-10-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.5429
Mathematics Subject Classification ID: 46B06
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 6354981
zbMATH Keywords: Banach spaces
zbMATH Keywords: asymptotic structure
zbMATH Keywords: spreading model
MaRDI profile type: Publication
OpenAlex ID: W2963400790
arXiv ID: 1310.5429
cites work: Q5317928
cites work: On the Structure of the Spreading Models of a Banach Space
cites work: Finite order spreading models
cites work: Higher order spreading models
cites work: OnB-convex Banach spaces
cites work: Q5337204
cites work: Q4839529
DOI: 10.4064/SM223-2-3
review text: Spreading models were established by \textit{A. Brunel} and \textit{L. Sucheston} [Math. Syst. Theory 7(1973), 294--299 (1974; Zbl 0323.46018)] and since their appearance they play a prominent role in the asymptotic theory of Banach spaces. Recently, \textit{S. A. Argyros} et al. [Fundam. Math. 221, No. 1, 23--68 (2013; Zbl 1296.46010)] provided a wide extension of the concept of a spreading model. In particular, given any Banach space \(X\) and any countable ordinal \(\xi\), they introduced the spreading models of order \(\xi\) associated to the space \(X\).NEWLINENEWLINEThe definition of the higher-order spreading models is based on the notion of \(\mathcal{F}\)-sequences and on plegma families. An \(\mathcal{F}\)-sequence in the space \(X\) is a sequence of the form \((x_s)_{s\in\mathcal{F}}\), where \(x_s\in X\) and the index set \(\mathcal{F}\) is a regular thin family of finite subsets of \(\mathbb{N}\). (A family of finite subsets of \(\mathbb{N}\) is called regular thin if it satisfies some specific properties. Any regular thin family \(\mathcal{F}\) is associated with a countable ordinal \(o(\mathcal{F})\) which measures the complexity of \(\mathcal{F}\) and is called the order of \(\mathcal{F}\).) A plegma family is a sequence \((s_1,s_2,\dots,s_l)\) of nonempty finite subsets of \(\mathbb{N}\) where the first elements of \(s_1,s_2,\dots,s_l\) are in increasing order and they lie before their second elements which are also in increasing order and so on. Now, given a sequence \((e_n)\) in a seminormed space \((E,\|\cdot\|_\ast)\), we say that \((e_n)\) is a spreading model of order \(\xi\) of the Banach space \(X\) if there are a regular thin family \(\mathcal{F}\) of order \(\xi\), an \(\mathcal{F}\)-sequence \((x_s)_{s\in\mathcal{F}}\) in \(X\) and an infinite subset \(M\) of \(\mathbb{N}\) such that for some null sequence \((\delta_n)_{n\in\mathbb{N}}\) of positive real numbers we have \(\left|\big\|\sum_{i=1}^k a_i x_{s_i}\big\|- \big\|\sum_{i=1}^k a_i …
reviewed by: Costas Poulios
description: scientific article; zbMATH DE number 6354981