Controlled Fusion Frames in Hilbert Spaces and Their Dual

Controlled frames in Hilbert spaces have been introduced by Balazs, Antoine and Grybos to improve the numerical output of in relation to algorithms for inverting the frame operator. In this paper, we introduce the notion of controlled fusion frames on Hilbert spaces. It is shown that controlled fusion frames are a generalization of fusion frames giving a generalized way to obtain numerical advantage in the sense of preconditioning to check the fusion frame condition. For this end, we introduce the notion of Q-duality for Controlled fusion frames. Also, we survey the robustness of controlled fusion frames under some perturbations.