Subfusion Frames

and Applied Analysis 3 where πVi is the orthogonal projection onto the subspace Vi. A fusion frame ν is called λ-tight fusion frame if C D λ, Parseval fusion frame if C D 1, and α-uniform fusion frame if α αi for all i ∈ I. If the second part of the above inequality is satisfied, then ν is called a Bessel fusion sequence for H with bound D. Similar to ordinary frames, the fusion frame operator Sν is defined by Sν ( f ) ∑ i∈I αi πV if, ∀f ∈ H. 2.3 Sν is a linear, positive, self-adjoint, and invertible operator and we have CI ≤ Sν ≤ DI. 2.4 A family of bounded operators {Ti}i∈I on H is called a resolution of identity on H if f ∑ i∈I Tif , for all f ∈ H. Proposition 2.3. Let { Vi, αi }i∈I be a fusion frame for H, Wi be a closed subspace of Vi and βi ≤ αi for all i ∈ I. Then { Wi, βi }i∈I is a Bessel fusion sequence for H. Proof. Since Wi is a closed subspace of Vi, πWiπVif πViπWif πWif and ‖πWif‖ ≤ ‖πVif‖ for all f ∈ H and for all i ∈ I. Hence,


Introduction
Frames were first introduced in 1924 by Duffin and Schaeffer 1 . Daubechies et al. in 2 found a fundamental new application. Nice properties of frames make them very useful in filter banks, sigma-delta quantization, signal and image processing. The theory of frames has been generalized rapidly, and various generalizations of frames have been proposed recently. Later general frame theory of subspaces was introduced by Casazza and Kutyniok 3 and Fornasier 4 as a natural generalization of the frame theory in Hilbert spaces. Since frames, particular frames of subspaces, are applied to signal processing, image processing, data compression, and sampling theory, we consider frames of subspaces on Hilbert spaces and extend some of the known results about bases and frames to frames of subspaces. Recently, the frames of subspaces have been renamed as fusion frames. This notion has been intensely studied earlier and several new applications have been discovered. The reader is referred to the works by Casazza and Kutyniok 3 and Gȃvruţa 5 . There exists a variety of applications which cannot be modeled naturally by one ordinary frame, for example, wireless sensor network 6 , sensor geophones in geophysics measurement and studies 7 , and the physiological structure of visual and hearing system 8 .
Let ν { V i , α i } i∈I be a fusion frame for H, W i be a closed subspace of V i and β i ≤ α i for all i ∈ I. If ω { W i , β i } i∈I is a fusion frame for H, then ω is called a subfusion frame of ν. If ν and ω are Bessel fusion sequences for H, then ω is called a Bessel subfusion sequence of ν.
In 9 , the authors introduced Bessel subfusion sequences and subfusion frames and they investigated the relationship between their operation. Also, the definition of the orthogonal complement of subfusion frames and the definition of the completion of Bessel fusion sequences were provided, and several results related with these notions were shown.
A notion related to subfusion frames has been brought in 10 , which is called frame of subspaces refinement shortly: Therefore, an FSR is a special subfusion frame and the authors have studied the excess of FSR in 10 .
In the present paper, we study the relations between fusion frames and subfusion frame operators. We also obtain some results about subfusion frame operators that these results are not true for fusion frame operators.
This paper is organized as follows. Section 2 briefly review the concept of frames, subfusion frames, and their properties. Section 3 includes some results of operator obtained of Bessel subfusion sequences. In 11 , the authors tried to show that the frame operator for a pair of fusion frames is bounded below and invertible, but we show this is not true. We further prove that the frame operator for a pair of subfusion frames is bounded below and invertible. Also, we will study operators for a pair of Bessel subfusion sequences. In Section 4, we study some constructions of subfusion frames. Finally, Section 5 contains a discussion on dual subfusion frames. In 5 , it has been shown that the dual fusion frame is a fusion frame. In this section, through an example, we show if ω Through this paper, H is a separable Hilbert space, I is a countable index set, and {V i } i∈I is a sequence of closed subspaces of H.

Review of Frames, Fusion Frames, and Subfusion Frames
In this section, we recall some definitions and basic properties of frames, fusion frames and subfusion frames. For more information, we refer the reader to 3, 9, 12, 13 . Definition 2.1. A sequence {f i } i∈I of elements in H is a frame for H if there exist positive constants A and B lower and upper frame bounds, resp. such that Let {V i } i∈I be a family of closed subspaces of a Hilbert space H and let {α i } i∈I be a family of weights, that is, α i > 0 for all i ∈ I. Then ν { V i , α i } i∈I is a fusion frame, if there exist positive constants C and D lower and upper fusion frame bounds, resp. such that Abstract and Applied Analysis 3 where π V i is the orthogonal projection onto the subspace V i . A fusion frame ν is called λ-tight fusion frame if C D λ, Parseval fusion frame if C D 1, and α-uniform fusion frame if α α i for all i ∈ I. If the second part of the above inequality is satisfied, then ν is called a Bessel fusion sequence for H with bound D. Similar to ordinary frames, the fusion frame operator S ν is defined by S ν is a linear, positive, self-adjoint, and invertible operator and we have There are examples such that { V i , α i } i∈I is a fusion frame, W i is a closed subspace of V i and β i ≤ α i for all i ∈ I, while { W i , α i } i∈I is not a fusion frame.
Since span i∈N

Operators between a Pair of Bessel Subfusion Sequences
In this section, we will study operators for a pair of Bessel subfusion sequences. Alternate dual frames and Bessel fusion sequences are important in the literature of frame theory because of their important role in applications. The notions of operators for a pair of Bessel fusion sequences and alternative dual of a fusion frame in H are defined by Gȃvruţa in 5 .
Let ν { V i , α i } i∈I and ω { W i , β i } i∈I be two Bessel fusion sequences for H. Then the frame operator for them is defined by 3.1 Moreover, if ν { V i , α i } i∈I is a fusion frame for H, with fusion frame operatore S ν , then ω is called an alternate dual of ν, if we have S ων is bounded and S ων S * νω . Recently, Khosravi and Musazadeh in 11, Proposition 2.9 tried to show that S ων is bounded below and invertible. "Let ν { V i , α i } i∈I be a fusion frame with fusion frame bounds C and D and fusion frame operator S ν for H. Let ω { W i , β i } i∈I be an alternate dual fusion frame for ν with required positivity. Then we have and also S ων is invertible." In the following example, we show that S ων is neither invertible nor bounded below.

3.5
For any f a, b, c in R 3 .
We have Now, let

3.10
Also S ων

6
Abstract and Applied Analysis Then S ων 3 3 0 3 3 0 0 0 1 . Therefore, S ων is not invertible. If f 1, −1, 0 , then there is not a positive number C such that Next, we show that under some conditions, S ων is invertible and bounded below. Proof. Let f be an arbitrary element of H. Then we have

3.15
Then S ων is invertible. Now, we are ready to describe the operator for a pair of subfusion frames. In this case, positivity, invertibility, and boundedness properties of these operators have been checked.

3.17
Then S ων is positive. Then we have where D is upper bound of ν. Hence,

3.20
Corollary 3.5. Let ω { W i , β i } i∈I be a Bessel subfusion sequence of ν { V i , α i } i∈I and S ων be invertible. Then ω is a subfusion frame of ν.
Proof. Since S ων is invertible, S ων is a below bounded. Then ω is a subfusion frame of ν. Proof.
Abstract and Applied Analysis then S ω ≤ S ων . We have Now we show that if ω be a subfusion frame of ν, then S ων is invertible.
then S ων S ω . Since S ω is invertible, hence, S ων is invertible. Proof. Let f be an arbitrary element of H. Then we have So S ων is bounded below, hence S ων is invertible and there exists a number B > 0 such that Then

Construction of Subfusion Frames
In this section, we study some new constructions of subfusion frames. Dealing with Bessel subfusion frames is important, since there are easy ways to turn such a family into subfusion frames. One way is to just add the subspace W 0 V 0 H to the families. We follow some ways that obtain new subfusion frames. Proof. By using the triangle inequality, for all f ∈ H, we have We define π V i : R 3 → V i by π V 1 a, b, c a, 0, 0 , π V 2 a, b, c π V 3 a, b, c a, b, c . { V i , α i } i∈I where I {1, 2, 3} is a fusion frame, We assume that W 1 e 1 , W 2 e 2 , W 3 R 3 . We define π W i : R 3 → W i by π W 1 a, b, c a, 0, 0 , π W 2 a, b, c 0, b, 0 , π W 3 a, b, c a, b, c .