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In this work, we start by introducing a general methodology to generate new relay fusion frames from given ones, namely the Spatial Complement Method, and analyze the relationships between the parameters of the original and the new relay fusion frame. We then present another simple approach to obtain relay fusion frames by considering fusion frames for its components. An explicit characterization concerning the existence of Parseval relay fusion frame consisting of two initial subspaces is given. Moreover, we obtain a necessary and sufficient condition under which the spatial complements of alternate dual relay fusion frames remain to be alternate dual relay fusion frames. Some results about Bessel relay fusion sequences are included. Finally, several examples are also given.

Some properties of controlled K-g-fusion frame have been discussed. Characterizations of controlled K-g-fusion frame are being presented. We also establish a relationship between quotient operator and controlled K-g-fusion frame. Some algebric properties of controlled K-g-fusion frame have been described. Finally, we shall discuss the stability of dual controlled g-fusion frame. Keywords: g-fusion frame, K-g-fusion frame, quotient operator, controlled g-fusion frame, controlled K-g-fusion frame. AMS Subject Classification: 42C15; 94A12; 46C07.

-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled -fusion frames, and we develop some results on the controlled -fusion frames for Hilbert spaces, which generalize some well known results of controlled fusion frame case. Also we discuss some characterizations of controlled Bessel -fusion sequences and of controlled -fusion frames. Further, we analyze stability conditions of controlled -fusion frames under perturbation.

Frames for operators or K-frames were recently considered by Gavruta (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.

In this paper, we demonstrate that any signal exhibiting a certain sparse pattern can be recovered in a stable and resilient manner through the utilization of the fusion frame approach. The theoretical analysis highlights that the deviation of the approximate solution is effectively controlled. Furthermore, the adoption of different norms contributes to further reinforcing the guarantees of robustness and stability. Driven by the ideas of compressed sensing and fusion frames, we extend the setting to relay fusion frames. With the help of operator theory, we provide several recovery guarantee conditions based on the relay fusion frames. Finally, the relationship between relay fusion frames and compressed sensing is elucidated.

In this paper, we analyze the relationship between relay fusion frames and standard fusion frames in finite-dimensional real Hilbert spaces. We propose an optimal design method for tight relay fusion frames in the setting of orthogonal subspaces. Additionally, we prove the existence of non-trivial relay operators and establish stability results for both subspaces and relay operators, showing that small perturbations preserve the relay fusion frame property with frame bounds converging to the original ones. We also present a sufficient condition for constructing relay fusion frames from scaled operators of existing fusion frames and show that invertible relay operators induce fusion frames.

In this paper, we introduce and investigate the concept of K-g-fusion frames in the Cartesian product of two Hilbert C*-modules over the same unital C*-algebra. Our main result establishes that the Cartesian product of two K-g-fusion frames remains a K-g-fusion frame for the direct-sum module. We give explicit formulae for the associated synthesis, analysis, and frame operators and prove natural relations (direct-sum decomposition of the frame operator). Furthermore, we prove a perturbation theorem showing that small perturbations of the component families, measured in the operator or norm sense, still yield a K-g-fusion frame for the product module, with explicit new frame bounds obtained.

In this paper, we present some new equalities and inequalities for K-g-fusion frames in Hilbert spaces with the help of operator theory. Our results generalize and improve the remarkable results which have been obtained by Ahmadi et al.

The relay fusion frames have been recently introduced in Hilbert spaces to model sensor relay networks and distributed sensor relay systems, which are deeply connected with compressed sensing. In this article, we introduce the notions of relay fusion frames and Banach relay fusion frames in Banach spaces and study certain attractive properties of relay fusion frames in this more general setting. In a particular sense, Schauder frames can be shown to be a special case of relay fusion frames. Moreover, the stability issue of relay fusion frames and Banach relay fusion frames will be addressed.

In this note, we give a new one-sided inequality for fusion frames in Hilbert C*-modules, which corrects one corresponding result. We also present some double inequalities for fusion frames in Hilbert C*-modules, which, compared to previous ones on this topic, possess different structures.