Abstract. A K-g-frame is a generalization of a g-frame. It can be used to reconstruct elements from the range of a bounded linear operator K in Hilbert spaces. K-g-frames have a certain advantage compared with g-frames in practical applications. In this paper, the interchangeability of two g-Bessel sequences with respect to a K-g-frame, which is different from a g-frame, is discussed. Several construction methods of K-g-frames are also proposed. Finally, by means of the methods and techniques in frame theory, several results of the stability of K-g-frames are obtained.

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G-frames and g-Riesz frames as generalized frames in Hilbert spaces have been studied by many authors in recent years. The super Hilbert space has a certain advantage compared with the Hilbert space in the field of studying quantum mechanics. In this paper, for super Hilbert spaceH⊕K, the definitions of a g-Riesz frame and minimal g-complete are put forward; also a characterization of g-Riesz frames is obtained. In particular, we generalize them to general super Hilbert spaceL1⊕L2⊕⋯⊕Ln. Finally, a conclusion of the stability of a g-Riesz frame for the super Hilbert space is given.

K-g-frame is a generalization ofg-frame. We generalize the tightg-frame toK-g-frame via atomic systems. In this paper, the definition of tightK-g-frame is put forward; equivalent characterizations and necessary conditions of tightK-g-frame are given. In particular, the necessary and sufficient condition for tightK-g-frame being tightg-frame is obtained. Finally, by means of methods and techniques of frame theory, several properties of tightK-g-frame are given.

K-frames were proposed recently as a generalization of frames. In this paper, fusing the ideas of K-frames and Riesz frames, the definitions of K-Riesz frames and K-Riesz bases for Hilbert space are put forward. Equivalent characterizations of K-Riesz frames and K-Riesz bases are also given. In particular, the K-Riesz frames's property which is different from Riesz frames is obtained. Finally, by means of methods and techniques of frame theory, several results of stability of K-Riesz frames and K-Riesz bases are obtained.