The correspondence of Fusion frames and frames In Hilbert $C^*$-modules and finite Gabor Fusion frames

Abstract

In this paper, we show that fusion frames in the finite dimensional Hilbert space $H$ correspond to frames in the Hilbert $C^*$-module $mathcal{B}left(mathbb{C}^nright)$. Moreover, we show that every tight fusion frame and Reisz fusion basis in $mathbb{C}^n$ correspond to a tight frame and Reisz basis in the Hilbert $C^*$-module $mathcal{B}left(mathbb{C}^nright)$ respectively. Then, we use this fact to characterize the dual of Reisz fusion basis. Finally, we introduce Gabor fusion frames as a new notion.