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 deals with frames of the form {T n ϕ} ∞ n=0 , where T ∈ B(H) belongs to certain classes of linear operators and ϕ ∈ H. The purpose of this paper is to investigate a new representation, namely, Fibonacci representation of sequences {fn} ∞ n=1 in a Hilbert space H; having the form fn+2 = T (fn + fn+1) for all n 1 and a linear operator T : span{fn} ∞ n=1 → span{fn} ∞ n=1 . We apply this kind of representations for complete sequences and frames. Finally, we present some properties of Fibonacci representation operators.

In this paper, we introduce pg-continuous frames and study the operators associated with a give pg-continuous frames. We also show many useful properties with corresponding notions, pg-continuous frames and pg-frames in Banach space.

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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}.