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

Abstract

In this paper, we provide some generalization of the concept of fusion frames following that evaluate their representability via a linear operator in Hilbert $C*$-module. We assume that $\Upsilon _\xi$ is self-adjoint and $\Upsilon _\xi(\frak{N} _\xi)= \frak{N} _\xi$ for all $\xi \in \mathfrak{S}$, and show that if a $g-$fusion frame $\{(\frak{N} _\xi, \Upsilon _\xi)\}_{\xi \in \mathfrak{S}}$ is represented via a linear operator $\mathcal{T}$ on $\hbox{span} \{\frak{N} _\xi\}_{ \xi \in \mathfrak{S}}$, then $\mathcal{T}$ is bounded. Moreover, if $\{(\frak{N} _\xi, \Upsilon _\xi)\}_{\xi \in \mathfrak{S}}$ is a tight $g-$fusion frame, then $\Upsilon_\xi $ is not represented via an invertible linear operator on $\hbox{span}\{\frak{N} _\xi\}_{\xi \in \mathfrak{S}}$, 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.