Representations of $g$-fusion frames in Hilbert $C^{\ast }$-Modules

In this paper, we provide some generalization of the concept of fusion frames following that evaluate their representability via a linear operator in Hilbert $C\ast$-module. We assume that ${\mathrm{{\rm Y}}}_{\xi }$ is self-adjoint and ${\mathrm{{\rm Y}}}_{\xi }({\mathfrak{N}}_{\xi })={\mathfrak{N}}_{\xi }$ for all $\xi \in \mathfrak{S}$, and show that if a $g-$fusion frame $\{({\mathfrak{N}}_{\xi },{\mathrm{{\rm Y}}}_{\xi }){\}}_{\xi \in \mathfrak{S}}$ is represented via a linear operator $\mathcal{T}$ on $\text{span}\{{\mathfrak{N}}_{\xi }{\}}_{\xi \in \mathfrak{S}}$, then $\mathcal{T}$ is bounded. Moreover, if $\{({\mathfrak{N}}_{\xi },{\mathrm{{\rm Y}}}_{\xi }){\}}_{\xi \in \mathfrak{S}}$ is a tight $g-$fusion frame, then ${\mathrm{{\rm Y}}}_{\xi }$ is not represented via an invertible linear operator on $\text{span}\{{\mathfrak{N}}_{\xi }{\}}_{\xi \in \mathfrak{S}}$, We show that, under certain conditions, a linear operator may also be used to express the perturbation of representable fusion frames. Finally, we’ll investigate the stability of this fusion frame type.