  Due to the importance of frame representation by a bounded operator in
dynamical sampling, researchers studied the frames of the form $\{T^{i-1}
f\}_{i\in \mathbb{N}}$, which $f$ belongs to separable Hilbert space
$\mathcal{H}$ and $T\in B(\mathcal{H})$, 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 $\{\Lambda T^{i-1} \in
B(\mathcal{H},\mathcal{K}):i\in \mathbb{N}\}$.
