After introducing [Formula: see text]-frames and fusion frames by Sun and Casazza, respectively, combining these frames together is an interesting topic for research. In this paper, we introduce the generalized fusion frames or [Formula: see text]-fusion frames for Hilbert spaces and give characterizations of these frames from the viewpoint of closed range and [Formula: see text]-fusion frame sequences. Also, the canonical dual [Formula: see text]-fusion frames are presented and we introduce a Parseval [Formula: see text]-fusion frame.

Frames for operators or K-frames were recently considered by Gȃvruţa (2012) in connection with atomic systems. Also, generalized frames are important frames in the Hilbert space of bounded linear operators. Fusion frames, which are a special case of generalized frames have various applications. This paper introduces the concept of generalized fusion frames for operators also known as K-g-fusion frames and we get some results for characterization of these frames. We further discuss dual and Q-dual in connection with K-g-fusion frames. Also we obtain some useful identities for these frames. We also give several methods to construct K-g-fusion frames. The results of this paper can be used in sampling theory which are developed by g-frames and especially fusion frames. In the end, we discuss the stability of a more general perturbation for K-g-fusion frames.

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The focus of this paper is mainly on the frames of operators or K-frames on Hilbert spaces in Parseval cases. Since equal-norm tight frames play an important role for transmitting robust data, we aim to study this topic on Parseval K-frames. We find that each finite set of equal-norm of vectors can be extended to an equal-norm K-frame. We also find a correspondence between Parseval K-frames and the set of all closed subspaces of a finite Hilbert space. Furthermore, we provide a construction of dual equal-norm K-frames.

In this paper, we first introduce the notation of weaving continuous fusion frames in separable Hilbert spaces. After reviewing the conditions for maintaining the weaving [Formula: see text]-fusion frames under the bounded linear operator and also, removing vectors from these frames, we will present a necessarily and sufficient condition about [Formula: see text]-woven and [Formula: see text]-fusion woven. Finally, perturbation of these frames will be introduced.