Robustness of controlled $K$-Fusion Frame in Hilbert C$^*$-modules under erasures of submodules

Controlled $\ast$-K-fusion frames are generalization of controlled fusion frames in frame theory. In this paper, we propose the notion of controlled $\ast$-k-fusions frames on Hilbert $C^{\ast}$-modules. We give some caraterizations and some of their properties are obtained. Then we study the erasures of submodules of a controlled $k$-fusion frame in Hilbert $C^{\ast}$-modules and we present some sufficient conditions under which a sequence remains a standart controlled k-fusion frame after deletion of some submodules. Finally, we introduce a perturbation for controlled $K$-fusion frames in Hilbert $C^{\ast}$-modules and it is shown that under some conditions controlled $K$-fusion frames are stable under this perturbation, and we generalize some of the results obtained for perturbations of controlled $K$-fusion frames.