The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form {T i f 1 } i∈N , where f 1 belongs to Hilbert space H and T : H → H belongs to certain classes of the bounded operators. Christensen et al., study frames for H with index set N (or Z), that have representations in the formAs frames of subspaces, fusion frames and generalized translation invariant systems are the spacial cases of g-frames, the purpose of this paper is to study gframes Λ = {Λ i ∈ B(H, K) : i ∈ I} (I = N or Z) having the form Λ i+1 = Λ 1 T i , for T ∈ B(H).

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Due to the importance of frame representation by a bounded operator in dynamical sampling, researchers studied the frames of the form {T i−1 f } i∈N , which f belongs to separable Hilbert space H and T ∈ B(H), and investigated the properties of T . Given that g-frames include the wide range of frames such as fusion frames, the main purpose of this paper is to study the characteristics of the operator T for g-frames of the form {ΛT i−1 ∈ B(H, K) : i ∈ N}.

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