Narrow Results

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces.

We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

In this paper, we develop the basic stability theory for g-frames and g-Riesz bases in Hilbert C*-module. Also, we study stability of the dual g-frames in Hilbert C*-module. We extend the Casazza–Christensen perturbation theorem to g-frames in Hilbert C*-module but this is not valid for g-Riesz bases. We prove some characterizations of g-frames and g-Riesz bases in Hilbert C*-module.

Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113] defined fusion frames in Hilbert spaces to split a large frame system into a set of (overlapping) much smaller systems and being able to process the data effectively locally within each sub-system. In this paper, we handle this problem using block sequences and generalized block sequences with respect to g-frames. Examples have been given to show their existence. A necessary and sufficient condition for a block sequence with respect to a g-frame to be a g-frame has been given. Finally, a sufficient condition for a generalized block sequence with respect to a g-frame to be a g-frame has been given.

In this paper, we study the erasures of submodules of a fusion frame in Hilbert $C^{\ast }$-modules and we present some sufficient conditions under which a sequence remains a standard fusion frame after deletion of some submodules. We also introduce a new perturbation for g-frames and fusion frames in Hilbert $C^{\ast }$-modules and it is shown that under some conditions g-frames and fusion frames are stable under this perturbation. We generalize some of the results obtained for perturbations of g-frames and fusion frames, especially we consider perturbations of direct sums, duals and approximate duals of g-frames.

Recently, Rashidi-Kouchi et al. devoted their efforts to giving analogs of the Casazza–Christensen general perturbation theorem of Hilbert space frames for the case of g-frames in Hilbert C^*-modules. We found, however, that in the proof they did not obtain the lower bound inequality of g-frames in Hilbert C^*-modules. In this paper, we establish an equivalent definition of g-frames in Hilbert C^*-modules, using it we present a complete proof for the stability result given by them.

For Hilbert spaces $H$ and $K_i(i\in 𝕁)$, we use the notations $B_H^g({\{K_j\}}_{j\in 𝕁})$, $F_H^g({\{K_j\}}_{j\in 𝕁})$ and $R_H^g({\{K_j\}}_{j\in 𝕁})$ to denote the sets of all $g$-Bessel sequences, $g$-frames and Riesz bases in $H$ with respect to ${\{K_j\}}_{j\in 𝕁}$, respectively. By defining a linear operation and a norm, we prove that $B_H^g({\{K_j\}}_{j\in 𝕁})$ becomes a Banach space and is isometrically isomorphic to the operator space $B(H,K)$, where $K={\oplus }_{j\in 𝕁}{K}_j$. In light of operator theory, it is proved that $F_H^g({\{K_j\}}_{j\in 𝕁})$ and $R_H^g({\{K_j\}}_{j\in 𝕁})$ are open sets in $B_H^g({\{K_j\}}_{j\in 𝕁})$. This implies that both $g$-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection $\pi$ from $B_H^g({\{K_j\}}_{j\in 𝕁})$ onto the $C^{\ast }$-algebra $B(H)$, a multiplication and an involution on the Banach space $B_H^g({\{K_j\}}_{j\in 𝕁})$ are defined so that $B_H^g({\{K_j\}}_{j\in 𝕁})$ becomes a unital $C^{\ast }$-algebra that is isometrically isomorphic to the $C^{\ast }$-algebra $B(H)$, provided that $H$ and $K$ are isomorphic.

In this paper, we use a new general sequence corresponding to a $g$-Bessel sequence $\{{\mathrm{\Lambda }}_j:j\in J\}$ to characterize that $\{{\mathrm{\Lambda }}_j:j\in J\}$ are $l^2({\{{{𝒱}}_j\}}_{j\in J})$-linear independent, $g$-complete and a $g$-frame. We also use $\{{\mathrm{\Lambda }}_j:j\in J\}$ and the refinement $\{{\mathrm{\Gamma }}_{ij}{\mathrm{\Lambda }}_i:i\in I,j\in J_i\}$ to characterize each other to be (near) exact $g$-frames or $g$-Riesz bases. Finally, we give several constructions and an equivalent characterization of Besselian $g$-frames and near exact $g$-frames.

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form ${\{T^{i-1}{f}_1\}}_{i\in \mathbb{N}}$, where $f_1$ belongs to Hilbert space ${\mathscr{H}}$ and $T:{\mathscr{H}}\to {\mathscr{H}}$ belongs to certain classes of bounded operators. Christensen et al. studied frames for ${\mathscr{H}}$ with index set $\mathbb{N}$ (or $\mathbb{Z}$), that has representations in the form ${\{T^{i-1}{f}_1\}}_{i\in \mathbb{N}}$ (or ${\{T^i{f}_0\}}_{i\in \mathbb{Z}}$). As frames of subspaces, fusion frames and generalized translation invariant systems are the special cases of $g$-frames, the purpose of this paper is to study and get sufficient conditions for $g$-frames $\mathrm{\Lambda }=\{{\mathrm{\Lambda }}_i\in B({\mathscr{H}},{𝒦}):i\in \mathbb{N}$ (or $\mathbb{Z})\}$ having the form ${\mathrm{\Lambda }}_{i+1}={\mathrm{\Lambda }}_1{T}^i,$$T\in B({\mathscr{H}})$ (or ${\mathrm{\Lambda }}_{i+1}={\mathrm{\Lambda }}_0{T}^i,$$T\in GL({\mathscr{H}})$). Also, we get the relation between representations of dual $g$-frames with index set $\mathbb{Z}$. Finally, we study stability of $g$-frame representations under some perturbations.