Tirilman 空间的对偶具有唯一的次对称基序列

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

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 positioning 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 t…

On uniqueness and plentitude of subsymmetric sequences

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization $Su(T^*)$ of Tsirelson's original Banach space provides the first known example of a space with a unique subsymmetric basic sequence that is additionally non-symmetric. Contrastingly, we provide a criterion for a space with a subsymmetric basis to contain a continuum of nonequivalent subsymmetric basic sequences and apply it to $Su(T^*)^*$. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some $\ell_p$ or $c_0$…

Homogeneous families on trees and subsymmetric basic sequences

We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $κ$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $κ$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic struc…

关于变指数勒贝格空间基本序列结构的研究

On the basic sequence structure of variable exponent Lebesgue spaces

本文研究了变指数勒贝格空间中子对称基本序列的结构，证明了在特定条件下此类序列与$\ell_r$标准基等价。

Baernstein 空间与 Schreier 空间上的严格奇异算子

Strictly singular operators on the Baernstein and Schreier spaces

该文证明在 Baernstein 与 Schreier 空间上，两个严格奇异算子复合为紧算子，并刻画了次对称基的等价性及非补块序列的存在。