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 sub-symmetric 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 c₀.