Generalization And Research Of Generalized Frames In Hilbert Spaces

Frame concept was introduced by Duffin and Schaeffer,have more advantages than these in practical applications.With the deepening research on frames,many generalized frames have been proposed,namely generalized frames,fusion frames,generalized fusion frames,K-g-fusion frames,controlled generalized frames and woven g-frames,etc.Generalized frames have more extensive properties and applications than classical frames.In this paper,the following research is done on the two generalizations about generalized frames,K-g-fusion frames and woven controlled g-frames.(1)Starting from known g-fusion Bessel sequences and K-g-fusion frame operators,the constructions and characterizations of K-g-fusion frames are studied.New K-g-fusion frames are constructed by utilizing operators with special properties.Next,we show a property that direct sum of K-g-fusion frames is also a K-g-fusion frame.Based on frame theorems,several results of stability and redundancy of K-g-fusion frames are obtained.(2)Combining the definitions of controlled g-frames and woven g-frames,the concept of woven controlled g-frames are put forward in Hilbert spaces.Based on frame theorems and operator theorems,characterizations and constructions of woven controlled g-frames.Finally,we show a property that direct sum of woven controlled g-frames is a woven controlled g-frame.