G-super Frames

Frames can provide a redundant and robust representation for a signal, and have been widely applied to many fields, such as filter bank theory, communications, signal and image processing. In2005, Sun introduced a natural generalization of frames, and proposed the definition of g-frames such as frames of subspaces, fusion frames, pseudo-frames and oblique frames. Many properties of g-frames can be shared with frames, but there also exist some difference between g-frames and frames. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases.Frames on super Hilbert spaces have been widely used in mobile communication network, satellite communication network and computer network, while g-frames on super Hilbert spaces are not reported. In this paper, we introduced the definition of g-frames on super Hilbert spaces——g-super frames, and discuss the characterization of (exact)g-frames and the stability of a (an exact)g-super frames under perturbation. The main work is as follows:1. We give the characterization of exact g-super frames and g-super Riesz bases;2. We consider the relationship between exact, g-super frames and g-super Riesz bases, and obtain an equivalent relation between an exact g-super frame and a g-super Riesz bases;3.We provide the stability of a(an exact) g-super frame under perturbation.