p-Frames in Separable Banach Spaces

Let X be a separable Banach space with dual X*. A countable family of elements {gi}⊂X* is a p-frame (1 p ∞) if the norm ‖⋅‖X is equivalent to the ℓp-norm of the sequence {gi(⋅)}. Without further assumptions, we prove that a p-frame allows every g∈X* to be represented as an unconditionally convergent series g=∑digi for coefficients {di}∈ℓq, where 1/p+1/q=1. A p-frame {gi} is not necessarily linear independent, so {gi} is some kind of “overcomplete basis” for X*. We prove that a q-Riesz basis for X* is a p-frame for X and that the associated coefficient functionals {fi} constitutes a p-Riesz basis allowing us to expand every f∈X (respectively g∈X*) as f=∑gi(f)fi (respectively g=∑g(fi)gi). In the general case of a p-frame such expansions are only possible under extra assumptions.