Frames for Banach Spaces

We use several fundamental results which characterize frames for a Hilbert space to give natural generalizations of Hilbert space frames to general Banach spaces. However, we will see that all of these natural generalizations (as well as the currently used generalizations) are equivalent to properties already extensively developed in Banach space theory. We show that the dilation characterization of framing pairs for a Hilbert spaces generalizes (with much more effort) to the Banach space setting. Finally, we examine the relationship between frames for Banach spaces and various forms of the Banach space approximation properties. We also consider Hilbert space frames. Here we classify the alternate dual frames for a Hilbert space frame by a natural manifold of operators on the Hilbert space. Most of the basic properties of alternate dual frames follow immediately from this characterization. We also answer a question of Han and Larson by showing that the dual frame pairs are compressions of Riesz basis and their dual bases. We also show that a family of frames for the same Hilbert space have the simultneous dilation property if and only if they have the same deficiency.