GENERALIZATION OF THE FRAME OPERATOR AND THE CANONICAL DUAL FRAME TO BANACH SPACES

X-d-frames for Banach spaces are generalization of Hilbert frames. In this paper we extend the concepts of frame operator and canonical dual to the case of X-d-frames. For a given X-d-frame {g(i)} for the Banach space X we define an X-d-frame map S : X -> X* and determine conditions, which imply that S is invertible and the family {S(-1)g(i)} is an X-d*-frame for X* such that f = Sigma g(i)(f)S(-1)g(i) for every f is an element of X and g = Sigma g(S(-1)g(i))g(i) for H every g is an element of X* If X is a Hilbert space and {g(i)} is a frame for X, then the l(2)-frame map S gives the frame operator S and the family {S(-1)g(i)} coincides with the canonical dual of {g(i)}.