g-frame representations with bounded operators

Abstract

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form $\{T^i f_1\}_{i\in \mathbb{N}}$, where $f_1$ belongs to Hilbert space $\h$ and $T:\h\rightarrow\h$ belongs to certain classes of the bounded operators. Christensen et al., study frames for $\h$ with index set $\mathbb{N}$ (or $\mathbb{Z}$), that have representations in the form $\{T^{i-1}f_1\}_{i\in \mathbb{N}}$ (or $\{T^if_0\}_{i\in \mathbb{Z}}$). As 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 $g$-frames $\Lambda=\{\Lambda_i\in B(\h,\K): i\in I\}$ $(I=\mathbb{N}$ or $\mathbb{Z}$) having the form $\Lambda_{i+1}=\Lambda_1 T^{i},$ for $T\in B(\h).$