Erasures and perturbations of g-frames and fusion frames in Hilbert \(C^\ast\)-modules (Q2800844)

The notion of g-frames and fusion frames in Hilbert \(C^\ast\)-modules are introduced by \textit{A. Khosravi} and \textit{B. Khosravi} [Int. J. Wavelets Multiresolut. Inf. Process. 6, No. 3, 433--446 (2008; Zbl 1153.46035)]. Perturbations of g-frames in Hilbert \(C^\ast\)-modules are studied by \textit{M. Rashidi-Kouchi} et al. in [Int. J. Wavelets Multiresolut. Inf. Process. 12, No. 6, Article ID 1450036, 16 p. (2014; Zbl 1317.42029)]. Fusion frames (frame of subspaces) are defined in [\textit{P. Casazza} and \textit{G. Kutyniok} Contemp. Math. 345, 87--113 (2004; Zbl 1058.42019 )] and g-frames are introduced in [\textit{W. Sun} J. Math. Anal. Appl. 322, No. 1, 437--452 (2006; Zbl 1129.42017)] on Hilbert spaces. In this paper, the author generalized the erasures of subspaces for fusion frames in Hilbert space from [\textit{P. Casazza} and \textit{G. Kutyniok} Contemp. Math. 464, 149--160 (2008; Zbl 1256.94017)] to Hilbert \(C^\ast\)-module setting. Then, he studied perturbations of g-frames and fusion frames in Hilbert \(C^\ast\)-modules such that Proposition 4.3 and Corollary 4.4 in [\textit{A. Khosravi} and \textit{K. Musazadeh} J. Math. Anal. Appl. 342, No. 2, 1068--1083 (2008; Zbl 1143.42033)] are consequences of this study. Finally, approximate duals of g-frames and their stability in Hilbert \(C^\ast\)-module have been investigated. These results generalize approximate duality of g-frames in Hilbert space introduced by the author and Khosravi in [\textit{A. Khosravi} and \textit{M. Mirzaee Azandaryani} Acta Math. Sci. 34, No. 2, 639--652 (2014; Zbl 1313.42089)].