The structure of symmetric basic sequences with applications to a class of Orlicz sequence spaces

We prove the following general result: Let  (xn)(xn)be a boundedly complete symmetric basis for a Banach space X. Then, for every symmetric basic sequence in X, we have the following alternatives: (a) it is equivalent to a basic sequence generated by a vector with respect to  (xn)(xn), or (b) it dominates a normalized block basis of  (xn)(xn)having coefficients tending to zero. This is an extension of a similar result obtained in 1973 by Altshuler, Casazza and Lin [1] for Lorentz sequence spaces. As an application, we obtain that, if M is a geometrically convex Orlicz function, then every symmetric basic sequence in the Orlicz sequence space  ℓMℓMhas the property (a) above, or it is equivalent to the standard basis of an  ℓpℓp-space.