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 ℓ p or c 0 .