Properties Of Fusion Frames And G-frames In Hilbert Spaces

The concept of frame was introduced by Duffin and Schaeffer to study nonharmonicFourier series in1952. In recent years, the theory of frame has got a series of results. Withthe development of the theory of frame, many authors proposed many generalized frames.The fusion frame and g-frame promote the study of frames. In this paper, we will makefurther research on fusion frames and g-frames.We discuss a relationship between a g-Riesz decomposition and a g-Riesz basis and weuse the operator theory to discuss the stability of a g-Riesz decomposition. Then we give anew g-Riesz decomposition. Fusion-Riesz frames, which were proposed recently asgeneralized Riesz frames in Hilbert spaces, so we discuss the properties of fusion-Rieszframes. We give the counterexample to show that not all fusion-Riesz frames have thesimilar characterization as Riesz frames, then we give the characterization of strongfusion-Riesz frame systems. And we show that strong fusion-Riesz frame systems andfusion-Riesz frames are different. Lastly, we discuss the stability of strong fusion-Rieszframe systems, fusion-Riesz frames、fusion-Besselian frames、near fusion-Riesz basis andfusion frames.