Stability of dual g-fusion frames in Hilbert spaces

We give a characterization of K-g-fusion frames and discuss the stability of dual g-fusion frames. We also present a necessary and sufficient condition for a quotient operator to be bounded.


Introduction
Frames in Hilbert spaces have many remarkable properties which make them very useful in processing of signals and images, filter bank theory, coding and communications, system modeling and many other fields. The notion of frame in Hilbert space was born in 1952 in the work of Duffin and Schaeffer [3] and their idea did not appear to make much general interest outside of non-harmonic Fourier series. Later on, after some innovative work of Daubechies, Grossman, Meyer [4], the theory of frames began to be studied more widely.
The theory of frames has been generalized rapidly and various generalizations of frames in Hilbert spaces namely, K-frames, G-frames, fusion frames etc. have been introduced in recent times. K-frames in Hilbert spaces were introduced by L. Gavruta [6] to study the atomic system relative to a bounded linear operator. Ramu and Johnson [9] obtained characterizations connecting K-frames and quotient operators. Sun [10] introduced a g-frame and a g-Riesz basis in complex Hilbert space and discussed several properties of them. g-frames were also defined by Kaftal,Larson,Zhang ([8]). Huang [7] began to study K-g-frame by combining K-frame and g-frame. General frame theory of subspaces were introduced by P. Casazza and G. Kutynoik [2] as a natural generalization of the frame theory in Hilbert spaces. Fusion frames and K-frames are the special case of generalized frames. Construction of K-g-fusion frames and their dual were presented by Sadri and Rahimi [12] to generalize the theory of K-frame, fusion frame and g-frame. In the theorey of frames, the stability of a frames is very important concept. The stability of g-frames and their dual g-frames have been studied by W. Sun [11] and proved that if two g-frames are closed to each other, so their dual g-frames are also closed to each other.
In this paper, we study the stability of dual g-fusion frames and see that dual gfusion frames are stable under small perturbation. Also, we give a characterization of K-g-fusion frames and at the end, we establish that a quotient operator will be bounded if and only if a g-fusion frame becomes U K-g-fusion frame.
Throughout this paper, H is considered to be a separable Hilbert space with associated inner product · , · and I H is the identity operator on H. We denote the collection of all bounded linear operators from H 1 to H 2 by B ( H 1 , H 2 ), where H 1 , H 2 are two Hilbert spaces. In particular B ( H ) denote the space of all bounded linear operators on H. For T ∈ B ( H ), we denote N ( T ) and R ( T ) for null space and range of T , respectively. Also P V ∈ B ( H ) is the orthogonal projection onto a closed subspace V ⊂ H. I, J will denote countable index sets and { H j } j ∈ J is a sequence of Hilbert spaces. Define with inner product is given by Clearly l 2 { H j } j ∈ J is a Hilbert space with the pointwise operations [12].

Preliminaries
In this section, we briefly recall some necessary definitions and results that will be needed later.
The set S ( H ) of all self-adjoint operators on H is a partially ordered set with respect to the partial order ≤ which is defined as for for some positive constants A, B. The constants A and B are called frame bounds.
for some constants 0 < A ≤ B < ∞. The constants A and B are called the lower and upper bounds of g-fusion frame, respectively. If A = B then Λ is called tight g-fusion frame and if A = B = 1 then we say Λ is a Parseval g-fusion frame. If the family Λ satisfies then it is called a g-fusion Bessel sequence for H with a bound B.
for H. Then the operator T Λ : is called synthesis operator and the operator T * Λ : is called analysis operator. The operator S Λ : H → H defined by is called g-fusion frame operator. It can be easily verify that Furthermore, if Λ is a g-fusion frame with bounds A and B, then from (1), The operator S Λ is bounded, self-adjoint, positive and invertible. Now, according to the Theorem (2.2), we can write, A I H ≤ S Λ ≤ B I H and this gives for some constants 0 < A ≤ B < ∞. The constants A and B are called the lower and upper bounds of K-g-fusion frame, respectively. If A = B then Λ is called a tight K-g-fusion frame. If K = I H then Λ is a g-fusion frame and if K = I H and Λ j = P W j for any j ∈ J, then Λ is a fusion frame for H.
3 Some properties of K-g-fusion frames Also, U is an invertible bounded linear operator on H, so for any j ∈ J, U W j is closed in H. Now, for each f ∈ H, using Theorem (2.1), we obtain On the other hand, Therefore, Γ is a U K U * -g-fusion frame for H.
Now, for each f ∈ H, we have Also, for each f ∈ H, we have Thus, Λ is a U − 1 K U -g-fusion frame for H with bounds A U 2 and B U − 1 2 .
be a g-fusion frame for H with frame bounds A, B and S Λ be the associated g-fusion frame operator. Then Now, for each f ∈ H, using Theorem (2.1), we get On the other hand, Proof. Proof of this Corollary directly follows from that of the Theorem (3.3), by putting K = P V . If Proof. Since T and T j ( for each j ∈ J ) are invertible, so By Theorem (2.1), for each f ∈ H, we have On the other hand, for each f ∈ H, we have (7) ] Thus, Γ is a K-g-fusion frame with bounds m 2 A T − 1 − 2 and M 2 B T 2 .
Remark 3.6. We further notice that the g-fusion frame operator S Γ of Γ satisfies the followings (I) By (2), (II) Moreover, if K = I H , i . e, if Λ is a g-fusion frame then Γ is also g-fusion frame. Then S Γ is invertible on H and by Theorem (2.2), we can write Remark 3.7. Let us now denote U = T * S − 1 Γ T and for each j ∈ J, L j = T * j T j and ∆ j = L j Λ j P W j U , where T, T j , Λ and Γ are all as in the Theorem (3.5). Now it is easy to verify the following: for all j ∈ J, L j ∈ B ( H j ) and ∆ j ∈ B ( H, H j ).

(III)
U and L j are self-adjoint and invertible.
(IV) From (11), it can be obtained (V) For each j ∈ J, using (7), Theorem 3.8. Let Λ be a g-fusion frame for H with bounds A and B. Then Proof. For each f ∈ H, we have (12) ].
Since for all j ∈ J, L j is invertible so Therefore, ∆ is a g-fusion frame for H. Furthermore, for each f ∈ H, we have (10) ].
According to the preceding procedure, we also get This completes the proof.

Stability of dual g-fusion frame
We is called as the canonical dual g-fusion frame of Λ. For each f ∈ H, the frame operator S Λ • of Λ • is described by In this section, we shall study the stability of dual g-fusion frames and at the end, a necessary and sufficient condition for some K ∈ B ( H ), for some invertible operator U ∈ B ( H ), a quotient operator will be bounded if and only if g-fusion frame becomes U K-g-fusion frame.
holds for each f ∈ H and for some D > 0 then Proof. Let I be any finite subset of J. Then by Cauchy-Schwarz inequality for each be the corresponding canonical dual g-fusion frames for Λ and Γ, respectively. Then the following statements hold: holds for each f ∈ H and for some D > 0 then for all f ∈ H, holds for each f ∈ H and for some D > 0 then for all f ∈ H, Proof. (I) Let S Λ and S Γ be the corresponding g-fusion frame operators for Λ and Γ, then for each f ∈ H, we have Since Λ • and Γ • are canonical dual g-fusion frames of Λ and Γ, so Then for any f ∈ H, using Theorem (2.9) and the proof of Theorem (4.1), we get Therefore, On the other hand, Since Λ is a g-fusion frame, for f ∈ H, we have Also, by given condition, we obtain Now, by Minkowski inequality, for each f ∈ H, we have (16) and (17) This completes the proof of (I).
Proof of (II). Since S Λ − S Γ is self-adjoint so Therefore, Now, for each f ∈ H, we have Then for each f ∈ H, we have This completes the proof.
holds for each f ∈ H and for some D > 0 then for all f ∈ H, Proof. In this case, we also find that Then for each f ∈ H, Theorem 4.4. Let K ∈ B ( H ) and Λ = { ( W j , Λ j , v j ) } j ∈ J be a K-g-fusion frame for H with frame operator S Λ . Let U ∈ B ( H ) be an invertible operator on H. Then the following statements are equivalent: (I) Γ = U W j , Λ j P W j U * , v j j ∈ J is a U K-g-fusion frame.
(II) The quotient operator Proof. (I) ⇒ (II) Since Γ is a U K-g-fusion frame, ∃ A, B > 0 such that for all f ∈ H, we have