Construction of g-fusion frames in Hilbert spaces

Abstract

After introducing [Formula: see text]-frames and fusion frames by Sun and Casazza, respectively, combining these frames together is an interesting topic for research. In this paper, we introduce the generalized fusion frames or [Formula: see text]-fusion frames for Hilbert spaces and give characterizations of these frames from the viewpoint of closed range and [Formula: see text]-fusion frame sequences. Also, the canonical dual [Formula: see text]-fusion frames are presented and we introduce a Parseval [Formula: see text]-fusion frame.

Controlled generalized fusion frame in the tensor product of Hilbert spaces

“…If A = B, then Λ is called tight g-fusion frame, and if A = B = 1, we say that Λ is a Parseval g-fusion frame. If Λ satisfies only the right inequality in (1), it is called a g-fusion Bessel sequence with the bound B in H (see [17]).…”

“…Frame theory has been widely used in signal and image processing, filter bank theory, coding and communications, etc. Several generalizations of frames, namely, g-frames [18], fusion frames [1], g-fusion frames [17], etc., have been introduced in recent times. A generalized fusion frame is used to generalize the theory of fusion frame and g-frame.…”

“…Clearly (see also [17]), l 2 {H i } i∈I is a Hilbert space with respect to the above inner product. We can similarly define the space l 2 {K j } j∈J .…”

Constructions of K-g-fusion and their dual in Hilbert spaces

“…Frames for operators or K-frames have been introduced by Gȃvruţa in [17] to study the nature of atomic systems for a separable Hilbert space with respect to a bounded linear operator K. It is a well-known fact that K-frames are more general than the classical frames and due to higher generality of K-frames, many properties of frames may not hold for K-frames. Recently, we presented g-fusion frames in [22]. This paper presents K-g-fusion frames with respect to a bounded linear operator on a separable Hilbert space which are a generalization of g-fusion frames.…”

“…Definition 2. (g-fusion frame) [22]. Let W = {W j } j∈J be a collection of closed subspaces of H, {v j } j∈J be a family of weights, i.e.…”