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. |