PG-FRAMES IN BANACH SPACES

For extending the concepts of p-frame, frame for Banach spaces and atomic decomposition, we will define the concept of pg-frame and g-frame for Banach spaces, by which each f is an element of X (X is a Banach space) can be represented by an unconditionally convergent series f = Sigma(gi) Delta(i,) where {Delta(i)}(i is an element of J) is a pg-frame, {g(i)} is an element of(Sigma circle plus Y-i(*))(lq) and 1/p + 1/q = 1. In fact, a pg-frame {Delta(i)} is a kind of an overcomplete basis for X*. We also show that every separable Banach space X has a g-Banach frame with bounds equal to 1.