New Properties of Generalized Fusion Frames in Hilbert \(C^∗\)-Modules

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In this paper, we provide some generalizations of the concept of fusion frames following that evaluate their representability via a linear operator in Hilbert \(C^*\)-modules. 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 investigate the stability of this type of fusion frames.

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