On uniqueness and plentitude of subsymmetric sequences

Abstract

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization $Su(T^{\ast })$ 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 $Su(T^{\ast }{)}^{\ast }$. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some ${\ell }_p$ or $c_0$.

19 pages

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