Duality of $g$-Bessel sequences and some results about RIP $g$-‎frames

In this paper, first we develop the duality concept for $g$-Bessel sequences and Bessel fusion sequences in Hilbert spaces. We obtain some results about dual, pseudo-dual and approximate dual of frames and fusion frames. We also expand every $g$-Bessel sequence to a frame by summing some elements. We define the restricted isometry property for $g$-frames and generalize some results from (B. G. Bodmann et al, Fusion frames and the restricted isometry property, Num. Func. Anal. Optim. 33 (2012) 770-790) to $g$-frame situation. Finally we study the stability of $g$-frames under erasure of operators.