Some equalities and inequalities for g-Bessel sequences in Hilbert spaces (II)

Some equalities and inequalities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>g</mml:mi></mml:math>-Bessel sequences in Hilbert spaces

In this paper, we first introduce the notation of weaving continuous fusion frames in separable Hilbert spaces. After reviewing the conditions for maintaining the weaving [Formula: see text]-fusion frames under the bounded linear operator and also, removing vectors from these frames, we will present a necessarily and sufficient condition about [Formula: see text]-woven and [Formula: see text]-fusion woven. Finally, perturbation of these frames will be introduced.

Generalized frames (in short, $g$-frames) are a natural generalization of standard frames in separable Hilbert spaces. Motivated by the concept of weaving frames in separable Hilbert spaces by Bemrose, Casazza, Grochenig, Lammers and Lynch in the context of distributed signal processing, we study weaving properties of $g$-frames. Firstly, we present necessary and sufficient con\-ditions for weaving $g$-frames in Hilbert spaces. We extend some results of \cite Bemrose, Casazza, Grochenig, Lammers and Lynch, and Casazza and Lynch regarding conversion of standard weaving frames to $g$-weaving frames. Some Paley-Wiener type perturbation results for weaving $g$-frames are obtained. Finally, we give necessary and sufficient conditions for weaving $g$-Riesz bases.

The concept of frames in Hilbert spaces continues to play a very interesting role in many kinds of applications. In this paper, we study the notion of dual continuous K -g-frames in Hilbert spaces. Also, we establish some new properties.

Continuous frames and fusion frames were considered recently as generalizations of frames in Hilbert spaces. In this paper, for any continuous fusion frame, we obtain a new family of inequalities which are parametrized by a parameter $$\lambda \in \mathbb {R}$$ . By suitable choices of $$\lambda $$ , one obtains the previous results as special cases. Moreover, these new inequalities involve the expressions $$\langle S_Yh,h \rangle $$ , $$\Vert S_Yh\Vert $$ , etc., where $$S_Y$$ is a “truncated form” of the continuous fusion frame operator.

In this paper, we first introduce the notion of controlled weaving [Formula: see text]-[Formula: see text]-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving [Formula: see text]-[Formula: see text]-frames in separable Hilbert spaces. Also, a characterization of controlled weaving [Formula: see text]-[Formula: see text]-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled [Formula: see text]-[Formula: see text]-woven frames give ordinary weaving [Formula: see text]-frames and vice-versa.

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.

Recently, fusion frames and frames for operators were considered as generalizations of frames in Hilbert spaces. In this paper, we generalize some of the known results in frame theory to fusion frames related to a linear bounded operator K which we call K-fusion frames. We obtain new K-fusion frames by considering K-fusion frames with a class of bounded linear operators and construct new K-fusion frames from given ones. We also study the stability of K-fusion frames under small perturbations. We further give some characterizations of atomic systems with subspace sequences.

Fusion frames and g-frames

Disjointness of frames in Hilbert spaces is closely related with superframes in Hilbert spaces and it also plays an important role in construction of superframes and frames, which were introduced and studied by Han and Larson. \(G\)-frame is a generalization of frame in Hilbert spaces, which covers many recent generalizations of frame in Hilbert spaces. In this paper, we study the \(g\)-frames in Hilbert spaces. We focus on the characterizations of disjointness of \(g\)-frames and constructions of \(g\)-frames. All types of disjointness are firstly characterized in terms of disjointness of frames induced by \(g\)-frames, then are characterized in terms of certain orthogonal projections. Finally we use disjoint \(g\)-frames to construct \(g\)-frames.

Frame theory is recently an active research area in mathematics, computer science and engineering with many exciting applications in a variety of different fields. This theory has been generalized rapidly and various generalizations of frames in Hilbert spaces. In this papers we study the notion of dual continuous $K$-frames in Hilbert spaces. Also we etablish some new properties.

Fusion frames and g-frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, g-frames, Banach g-frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that g-frames, fusion frames and Banach g-frames are stable under small perturbations and invertible operators.

In this paper, we introduce and characterize controlled dual frames in Hilbert spaces. We also investigate the relation between bounds of controlled frames and their related frames. Then, we define the concept of approximate duality for controlled frames in Hilbert spaces. Next, we introduce multiplier operators of controlled frames in Hilbert spaces and investigate some of their properties. Finally, we show that the inverse of a controlled multiplier operator is also a controlled multiplier operator under some mild conditions.

Continuous Frames in Hilbert Space

Fusion frames are generalizations of frames in Hilbert spaces which were introduced by Casazza et al. (2008). In the present paper, we study the relations between fusion frames and subfusion frame operators. Specially, we introduce new construction of subfusion frames and derive new results.