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

TL;DR

This paper introduces controlled $ extit{K}$-fusion frames in Hilbert $C^*$-modules, explores their properties, robustness under submodule erasures, and stability under perturbations, extending frame theory in operator algebra contexts.

Contribution

It defines controlled $ extit{K}$-fusion frames on Hilbert $C^*$-modules, analyzes their properties, and establishes conditions for their robustness and stability under erasures and perturbations.

Findings

Controlled $ extit{K}$-fusion frames are characterized and their properties are established.

Sufficient conditions are provided for frames to remain stable after submodule deletions.

Controlled $ extit{K}$-fusion frames are shown to be stable under certain perturbations.

Abstract

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.