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

Xd-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 Xd-frames. For a given Xd-frame {gi} for the Banach space X we define an Xd-frame map𝕊 : X → X* and determine conditions, which imply that 𝕊 is invertible and the family {𝕊-1gi} is an [Formula: see text]-frame for X* such that f = ∑gi(f)𝕊-1gi for every f ∈ X and g = ∑g(𝕊-1gi)gi for every g ∈ X*. If X is a Hilbert space and {gi} is a frame for X, then the ℓ2-frame map 𝕊 gives the frame operator S and the family {𝕊-1gi} coincides with the canonical dual of {gi}.