p-Frames in Separable Banach Spaces

Abstract

Let X be a separable Banach space with dual X ^*. A countable family of elements {g _i }⊂X ^* is a p-frame (1 p ∞) if the norm ‖⋅‖ _X is equivalent to the ℓ^p-norm of the sequence {g _i (⋅)}. Without further assumptions, we prove that a p-frame allows every g∈X ^* to be represented as an unconditionally convergent series g=∑d _i g _i for coefficients {d _i }∈ℓ^q, where 1/p+1/q=1. A p-frame {g _i } is not necessarily linear independent, so {g _i } 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 {f _i } constitutes a p-Riesz basis allowing us to expand every f∈X (respectively g∈X ^*) as f=∑g _i (f)f _i (respectively g=∑g(f _i )g _i ). In the general case of a p-frame such expansions are only possible under extra assumptions.