On g-frame representations via linear operators

Let {Mk}k?Z be a sequence of closed subspaces of Hilbert space H, and let {Tk}k?Z be a sequence of linear operators from H into Mk, k ? Z. In the case where, Tk is selfadjoint andTk(Mk) = Mk for all k ? Z, we show that if a g-frame {(Mk, Tk)}k?Z is represented via a linear operator T on span{Mk}k?Z, then T is bounded; moreover, if {(Mk, Tk)}k?Z is a tight g- frame, then T is not invertible. We also study the perturbation and the stability of these g-frames. Finally, we give some examples to show the validity of the results. A preliminary version of this manuscript was submitted to https://arxiv.org/abs/2305.08182 This version is a reedited copy of it.