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 $Υ_ξ$ is self-adjoint and $Υ_ξ(\frak{N} _ξ)= \frak{N} _ξ$ for all $ξ\in \mathfrak{S}$, and show that if a $g-$fusion frame $\{(\frak{N} _ξ, Υ_ξ)\}_{ξ\in \mathfrak{S}}$ is represented via a linear operator $\mathcal{T}$ on $\hbox{span} \{\frak{N} _ξ\}_{ ξ\in \mathfrak{S}}$, then $\mathcal{T}$ is bounded. Moreover, if $\{(\frak{N} _ξ, Υ_ξ)\}_{ξ\in \mathfrak{S}}$ is a tight $g-$fusion frame, then $Υ_ξ$ is not represented via an invertible linear operator on $\hbox{span}\{\frak{N} _ξ\}_{ξ\in \mathfrak{S}}$, 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.