International Journal of Analysis and Applications

Volume 19, Number 6 (2021), 836-857

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-836

K −g−FUSION FRAMES IN HILBERT C∗−MODULES

FAKHR-DINE NHARI1,∗, RACHID ECHARGHAOUI1 AND MOHAMED ROSSAFI2

1Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty Of Sciences,

University of Ibn Tofail, Kenitra, Morocco

2LaSMA Laboratory Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi

Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco

∗Corresponding author: nharidoc@gmail.com

Abstract. In this paper, we introduce the concepts of g−fusion frame and K −g−fusion frame in Hilbert

C∗−modules and we give some properties. Also, we study the stability problem of g−fusion frame. The

presented results extend, generalize and improve many existing results in the literature.

1. Introduction and Preliminaries

Frame theory is recently an active research area in mathematics, computer science, and engineering with

many exciting applications in a variety of different fields.

A frame is a set of vectors in a Hilbert space that can be used to reconstruct each vector in the space from

its inner products with the frame vectors. These inner products are generally called the frame coefficients of

the vector. But unlike an orthonormal basis each vector may have infinitely many different representations

in terms of its frame coefficients.

Frames for Hilbert spaces were introduced by Duffin and Schaefer [1] in 1952 to study some deep problems

in nonharmonic Fourier series by abstracting the fundamental notion of Gabor [2] for signal processing.

Received September 29th, 2021; accepted October 21st, 2021; published October 28th, 2021.

2010 Mathematics Subject Classification. 42C15.

Key words and phrases. fusion frame; g-fusion frame; K − g−fusion; Hilbert C∗−module.
©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

836

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-836


Int. J. Anal. Appl. 19 (6) (2021) 837

Hilbert C∗-modules is a generalization of Hilbert spaces by allowing the inner product to take values in

a C∗-algebra rather than in the field of complex numbers.

Many generalizations of the concept of frame have been defined in Hilbert C∗-modules [3, 5, 9–13].

In the following, we recall some definitions and results that will be used to prove our mains results.

Let A be a unital C∗−algebra, let J be countable index set.

Throughout this paper H and K are countably generated Hilbert A−modules and (Kj)j∈J is a sequence

of closed Hilbert submodules of K.

For each j ∈ J, End∗A(H,Kj) is the collection of all adjointable A−linear maps from H to Kj, and

End∗A(H,H) is denoted by End
∗
A(H).

Definition 1.1. [6]

Let A be a unital C∗-algebra and H be a left A-module, such that the linear structures of A and H are

compatible. H is a pre-Hilbert A-module if H is equipped with an A-valued inner product 〈., .〉 : H×H→A,

such that is sesquilinear, positive definite and respects the module action. In the other words,

(i) 〈f,f〉≥ 0 for all f ∈H and 〈f,f〉 = 0 if and only if f = 0.

(ii) 〈af + g,h〉 = a〈f,h〉 + 〈g,h〉 for all a ∈A and f,g,h ∈H.

(iii) 〈f,g〉 = 〈g,f〉∗ for all f,g ∈H.

For f ∈H, we define ||f|| = ||〈f,f〉||
1
2 . If H is complete with ||.||, it is called a Hilbert A-module or a Hilbert

C∗-module over A. For every a in a C∗-algebra A, we have |a| = (a∗a)
1
2 and the A-valued norm on H is

defined by |f| = 〈f,f〉
1
2 for f ∈H.

Define l2((Kj)j∈J) by

l2((Kj)j∈J) = {(fj)j∈J : fj ∈Kj, ||
∑
j∈J
〈fj,fj〉|| < ∞}.

With A−valued inner product is given by

〈(fj)j∈J, (gj)j∈J〉 =
∑
j∈J
〈fj,gj〉,

l2((Kj)j∈J) is a Hilbert A−module.

Lemma 1.1. [7] Let T ∈ End∗A(H,K) and H,K are Hilberts A−modules.The following statemnts are

multually equivalent:

(i) T is surjective.

(ii) T∗ is bounded below with respect to the norm, i.e., there is m > 0 such that ||T∗f|| ≥ m||f|| for all

f ∈K.

(iii) T∗ is bounded below with respect to the inner product, i.e, there is m
′
> 0 such that 〈T∗f,T∗f〉 ≥

m
′
〈f,f〉 for all f ∈K.



Int. J. Anal. Appl. 19 (6) (2021) 838

Lemma 1.2. [7] Let H be a Hilbert A-module over a C∗-algebra A, and T ∈ End∗A(H) such that T
∗ = T .

The following statements are equivalent:

(i) T is surjective.

(ii) There are m,M > 0 such that m‖f‖≤‖Tf‖≤ M‖f‖, for all f ∈H.

(iii) There are m′,M′ > 0 such that m′〈f,f〉≤ 〈Tf,Tf〉≤ M′〈f,f〉 for all f ∈H.

Lemma 1.3. [4] Let H and K are two Hilbert A-modules and T ∈ End∗(H,K). Then:

(i) If T is injective and T has closed range, then the adjointable map T∗T is invertible and

‖(T∗T)−1‖−1 ≤ T∗T ≤‖T‖2.

(ii) If T is surjective, then the adjointable map TT∗ is invertible and

‖(TT∗)−1‖−1 ≤ TT∗ ≤‖T‖2.

Lemma 1.4. [8] Let H be a Hilbert A-module. If T ∈ End∗A(H), then

〈Tf,Tf〉≤ ‖T‖2〈f,f〉, ∀f ∈H.

Lemma 1.5. [14] Let A be a C∗−algebra, E, H, K be Hilbert A-modules. Let T ∈ End∗A(E,K) and

T
′
∈ End∗A(H,K). If R(T∗) is orthogonally complemented, then the following statements are equivalent:

(i) R(T
′
) ⊆R(T).

(ii) T
′
(T
′
)∗ ≤ λTT∗ for some λ > 0.

(iii) There exists a positive real number µ > 0 such that ‖(T
′
)∗f‖≤ µ‖T∗f‖, for all f ∈K.

(iv) There exists a solution X ∈ End∗A(H,E) of the so-called douglas equation T
′

= TX.

2. K −g−fusion frames in Hilbert C∗−modules

We begin this section with the following lemma.

Lemma 2.1. Let (Wj)j∈J be a sequence of orthogonally complemented closed submodules of H and T ∈

End∗A(H) invertible, if T
∗TWj ⊂ Wj for each j ∈ J, then (TWj)j∈J is a sequence of orthogonally comple-

mented closed submodules and PWjT
∗ = PWjT

∗PT Wj .

Proof. Firstly for each j ∈ J, T : Wj → TWj is invertible, so each TWj is a closed submodule of H. We

show that H = TWj ⊕T(W⊥j ). Since H = TH, then for each f ∈H, there exists g ∈H sutch that f = Tg.

On the other hand g = u + v, for some u ∈ Wj and v ∈ W⊥j . Hence f = Tu + Tv, where Tu ∈ TWj and

Tv ∈ T(W⊥j ) plainly TWj ∩T(W
⊥
j ) = (0), therefore H = TWj ⊕T(W

⊥
j ). Hence for every g ∈ Wj, h ∈ W

⊥
j

we have T∗Tg ∈ Wj and therefore 〈Tg,Th〉 = 〈T∗Tg,h〉 = 0, so T(W⊥j ) ⊂ (TWj)
⊥ and consequentely

T(W⊥j ) = (TWj)
⊥ witch implies that TWj is orthogonally complemented.



Int. J. Anal. Appl. 19 (6) (2021) 839

Let f ∈ H we have f = PT Wjf + g, for some g ∈ (TWj)⊥, then T∗f = T∗PT Wjf + T∗g. Let v ∈ Wj

then 〈T∗g,v〉 = 〈g,Tv〉 = 0 then T∗g ∈ W⊥j and we have PWjT
∗f = PWjT

∗PT Wjf + PWjT
∗g, then

PWjT
∗f = PWjT

∗PT Wjf thus implies that for each j ∈ J we have PWjT∗ = PWjT∗PT Wj . �

Now we define the notion of K −g−fusion frame in Hilbert C∗−modules.

Definition 2.1. let A be a unital C∗−algebra and H be a countably generated Hilbert A−module. let (vj)j∈J
be a family of weights in A,i.e.,each vj is a positive invertible element frome the center of A. Let (Wj)j∈J
be a collection of orthogonally complemented closed submodules of H. Let (Kj)j∈J a sequence of closed

submodules of K and Λj ∈ End∗A(H,Kj) for each j ∈ J. we say Λ = (Wj, Λj,vj)j∈J is g−fusion frame for

H with respect to (Kj)j∈J if there exist real constants 0 < A ≤ B < ∞ such that

A〈f,f〉≤
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B〈f,f〉, ∀f ∈H.

The counstants 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 ∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B〈f,f〉, ∀f ∈H.

Then it is called a g−fusion bessel sequence for H with bound B.

Lemma 2.2. let Λ = (Wj, Λj,vj)j∈J be a g−fusion bessel sequence for H with bound B. Then for each

sequence (fj)j∈J ∈ l2((Kj)j∈J), the series
∑

j∈J vjPWj Λ
∗
jfj is converge unconditionally.

Proof. let I be a finite subset of J, then

||
∑
j∈I

vjPWj Λ
∗
jfj|| = sup

||g||=1
||〈
∑
j∈I

vjPWj Λ
∗
jfj,g〉||

≤ ||
∑
j∈I
〈fj,fj〉||

1
2 sup
||g||=1

||
∑
j∈I

v2j〈ΛjPWjg, ΛjPWjg〉||
1
2

≤
√
B||

∑
j∈I
〈fj,fj〉||

1
2 .

And it follows that
∑

j∈J vjPWj Λ
∗
jfj is unconditionally convergent in H. �

Now, we can define the synthesis operator by lemma 2.2

Definition 2.2. let Λ = (Wj, Λj,vj)j∈J be a g−fusion bessel sequence for H. Then the operator TΛ :

l2((Kj)j∈J)) →H defined by

TΛ((fj)j∈J) =
∑
j∈J

vjPWj Λ
∗
jfj, ∀(fj)j∈J ∈ l

2((Kj)j∈J).



Int. J. Anal. Appl. 19 (6) (2021) 840

Is called synthesis operator. We say the adjoint T∗Λ of the synthesis the analysis operator and it is defined by

T∗Λ : H→ l
2((Kj)j∈J) such that

T∗Λ(f) = (vjΛjPWj (f))j∈J, ∀f ∈H.

The operator SΛ : H→H defined by

SΛf = TΛT∗Λf =
∑
j∈J

v2j PWj Λ
∗
j ΛjPWj (f), ∀f ∈H.

Is called g−fusion frame operator. It can be easily verify that

(2.1) 〈SΛf,f〉 =
∑
j∈J

v2j〈ΛjPWj (f), ΛjPWj (f)〉, ∀f ∈H.

Furthermore, if Λ is a g−fusion frame with bounds A and B, then

A〈f,f〉≤ 〈SΛf,f〉≤ B〈f,f〉, ∀f ∈H.

It easy to see that the operator SΛ is bounded, self-adjoint, positive, now we proof the inversibility of SΛ.

Let f ∈H we have

||T∗Λ(f)|| = ||(vjΛjPWj (f))j∈J|| = ||
∑
j∈J

v2j〈ΛjPWj (f), ΛjPWj (f)〉||
1
2 .

Since Λ is g−fusion frame then
√
A||〈f,f〉||

1
2 ≤ ||T∗Λf||.

Then
√
A||f|| ≤ ||T∗Λf||.

Frome lemma1.1, TΛ is surjective and by lemma1.3, TΛT
∗
Λ = SΛ is invertible. We now, AIH ≤ SΛ ≤ BIH

and this gives B−1IH ≤S−1Λ ≤ A
−1IH

Definition 2.3. Let A be a unital C∗−algebra and H be a countably generated Hilbert A−module. let (vj)j∈J
be a family of weights in A,i.e.,each vj is a positive invertible element frome the center of A, let (Wj)j∈J be a

collection of orthogonally complemented closed submodules of H. Let (Kj)j∈J a sequence of closed submodules

of K and Λj ∈ End∗A(H,Kj) for each j ∈ J and K ∈ End
∗
A(H). We say Λ = (Wj, Λj,vj)j∈J is K−g−fusion

frame for H with respect to (Kj)j∈J if there exist real constants 0 < A ≤ B < ∞ such that

(2.2) A〈K∗f,K∗f〉≤
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B〈f,f〉, ∀f ∈H.

The constants A and B are called a lower and upper bounds of K − g−fusion frame, respectively. If the

left-hand inequality of (2.2) is an equality, we say that Λ is a tight K −g−fusion frame. If K = IH then Λ

is a g−fusion frame and if K = IH and Λj = PWj for any j ∈ J, then Λ is a fusion frame for H



Int. J. Anal. Appl. 19 (6) (2021) 841

Example 2.1. Let H be a Hilbert C∗−module with dimensional 3 and let {e1,e2,e3} be standard basis.

We define the operator K on H by

Ke1 = e2, Ke2 = e3, Ke3 = e3;

Suppose that Wj = Kj = span{ej} where j = 1, 2, 3. Let

Λjx = 〈x,ej〉ej,

it is clear that (Wj, Λj, 1)j∈J is a K −g−fusion frame for H.

Remark 2.1. If Λ = (Wj, Λj,vj)j∈J is K −g−fusion frame for H with bounds A and B, then we have

(2.3) AKK∗ ≤SΛ ≤ BIH

From inequalities (2.1) and (2.3), we have

Lemma 2.3. Let K ∈ End∗A(H) and Λ = (Wj, Λj,vj)j∈J be a g−fusion bessel sequence for H. Then Λ is

K −g−fusion frame for H if and only if there exist a constant A > 0 such that AKK∗ ≤ SΛ, where SΛ is

the frame operator for Λ.

Theorem 2.1. Let K ∈ End∗A(H), and Λ = (Wj,PWj,vj)j∈J be a g−fusion bessel sequence for H with

frame operator SΛ such that R(S
1
2

Λ ) is orthogonally complemented. Then Λ is K −g−fusion frame for H if

and only if K = S
1
2

Λ M for some M ∈ End
∗
A(H).

Proof. Suppose Λ is a K −g−fusion frame for H, then there exist A > 0 such that AKK∗ ≤SΛ, and SΛ is

self-adjoint and positive thus S
1
2

Λ is self-adjoint and positive, so we have

KK∗ ≤
1

A
S

1
2

ΛS
1
2

Λ .

By lemma1.5, there exists some M ∈ End∗A(H) such that K = S
1
2

Λ M.

Suppose that there exists an operator M ∈ End∗A(H) so that K = S
1
2

Λ M. From Lemma1.5 , we know

that AKK∗ ≤SΛ for some constant A > 0, from Lemma2.3, Λ is a K −g−fusion frame. �

Theorem 2.2. If U ∈ End∗A(H) and Λ is a K−g−fusion frame for H, and R(U) ⊂R(K) such that R(K∗)

is orthogonaly complemented. Then Λ is U −g−fusion frame for H.

Proof. By Lemma1.5, ∃λ > 0: UU∗ ≤ λKK∗, then for each f ∈H we have

〈U∗f,U∗f〉 = 〈UU∗f,f〉≤ 〈λKK∗f,f〉≤ λ〈K∗f,K∗f〉.

It follows that,
A

λ
〈U∗f,U∗f〉≤

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉, ∀f ∈H.

So Λ is a U −g−fusion frame for H. �



Int. J. Anal. Appl. 19 (6) (2021) 842

Theorem 2.3. Let Λ = (Wj, Λj,vj)j∈J and Γ = (Vj, Γj,uj)j∈J be two g−fusion bessel sequence for H with

bounds B1 and B2, respectively. Suppose that TΛ and TΓ are their synthesis operators such that TΓT
∗
Λ = K

∗

where K ∈ End∗A(H). Then, both Λ and Γ are K and K
∗ −g−fusion frames, respectively.

Proof. Let f ∈H, we have

〈K∗f,K∗f〉 = 〈TΓT∗Λf,TΓT
∗
Λf〉≤ ||TΓ||

2〈T∗Λf,T
∗
Λf〉≤ B2

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉.

So,

B−12 〈K
∗f,K∗f〉≤

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉.

Thus Λ is K − g− fusion frame for H. Similarly, Γ is K∗ − g−fusion frame for H with the lower bound

B−11 . �

Theorem 2.4. Let K ∈ End∗A(H) and Λ = (Wj, Λj,vj)j∈J be a g−fusion bessel sequence for H, with

synthesis operator TΛ of Λ. Suppose that R(T∗Λ) and R(K∗) are orthogonaly complemented, the following

statements hold:

(1) If Λ is a tight K −g−fusion frame for H, then R(K) = R(TΛ).

(2) R(K) = R(TΛ) if and only if there exist two constants 0 < A ≤ B < ∞ such that:

A〈K∗f,K∗f〉≤
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B〈K
∗f,K∗f〉, ∀f ∈H.

Proof. (1) Suppose that Λ is a tight K −g−fusion frame for H, then there exixt A > 0, such that for each

f ∈H

A〈K∗f,K∗f〉 =
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉

= 〈(vjΛjPWjf)j∈J, (vjΛjPWjf)j∈J〉

= 〈T∗Λf,T
∗
Λf〉.

Then

〈AKK∗f,f〉 = 〈TΛT∗Λf,f〉.

So

AKK∗ = TΛT
∗
Λ.

Then by lemma1.5 , R(TΛ) = R(K)

(2) Suppose that R(K) = R(TΛ), by lemma1.5 there exist two constants A, B > 0 such that

AKK∗ ≤ TΛT∗Λ ≤ BKK
∗.



Int. J. Anal. Appl. 19 (6) (2021) 843

Which implies that for each f ∈H

〈AKK∗f,f〉≤ 〈TΛT∗Λf,f〉≤ 〈BKK
∗f,f〉.

A〈KK∗f,f〉≤ 〈TΛT∗Λf,f〉≤ B〈KK
∗f,f〉.

Therefore

A〈K∗f,K∗f〉≤
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B〈K
∗f,K∗f〉.

Suppose that there exist two constants A, B > 0 such that for each f ∈H

A〈K∗f,K∗f〉≤
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B〈K
∗f,K∗f〉.

Then

A〈KK∗f,f〉≤ 〈TΛT∗Λf,f〉≤ B〈KK
∗f,f〉.

So

AKK∗ ≤ TΛT∗Λ ≤ BKK
∗.

Since by lemma1.5 , R(TΛ) = R(K). �

Theorem 2.5. Let U ∈ End∗A(H) be an invertible operator on H and Λ = (Wj, Λj,vj)j∈J be a K−g−fusion

frame for H for some K ∈ End∗A(H). Suppose that U
∗UWj ⊂ Wj, ∀j ∈ J. Then Γ = (UWj, ΛjPWjU∗,vj)j∈J

is a UKU∗ −g−fusion frame for H.

Proof. Since Λ is a K −g−fusion frame for H, ∃ A,B > 0 such that

A〈K∗f,K∗f〉≤
∑
j∈J

v2j〈ΛjPWj (f), ΛjPWj (f)〉≤ B〈f,f〉, ∀f ∈H.

Also, U is an invertible linear operator on H, so for any j ∈ J, UWj is closed in H. Now, for each f ∈ H,

using lemma2.1, we obtain

∑
j∈J

v2j〈ΛjPWjU
∗PUWj (f), ΛjPWjU

∗PUWj (f)〉 =
∑
j∈J

v2j〈ΛjPWjU
∗(f), ΛjPWjU

∗(f)〉

≤ B〈U∗f,U∗f〉

≤ B||U||2〈f,f〉.



Int. J. Anal. Appl. 19 (6) (2021) 844

On the other hand, for each f ∈H

A〈(UKU∗)∗f, (UKU∗)∗f〉 = A〈UK∗U∗f,UK∗U∗f〉

≤ A||U||2〈K∗U∗f,K∗U∗f〉

≤ ||U||2
∑
j∈J

v2j〈ΛjPWjU
∗(f), ΛjPWjU

∗(f)〉

≤ ||U||2
∑
j∈J

v2j〈ΛjPWjU
∗PUWj (f), ΛjPWjU

∗PUWj (f)〉,

Then

A

||U||2
〈(UKU∗)∗f, (UKU∗)∗f〉≤

∑
j∈J

v2j〈ΛjPWjU
∗PUWj (f), ΛjPWjU

∗PUWj (f)〉

Therefore, Γ is UKU∗ −g−fusion frame for H. �

Theorem 2.6. Let U ∈ End∗A(H) be an invertible operator on H and Γ = (UWj, ΛjPWjU
∗,vj)j∈J be

a K − g−fusion frame for H for some K ∈ End∗A(H). Suppose that U
∗UWj ⊂ Wj, ∀j ∈ J. Then

Λ = (Wj, Λj,vj)j∈J is a U
−1KU −g−fusion frame for H.

Proof. Since Γ = (UWj, ΛjPWj,vj)j∈J is K −g−fusion frame for H, for all f ∈H, ∃ A, B > 0 such that

A〈K∗f,K∗f〉≤
∑
j∈J

v2j〈ΛjPWjU
∗PUWj, ΛjPWjU

∗PUWj〉≤ B〈f,f〉.

Let f ∈H, we have

A〈(U−1KU)∗f, (U−1KU)∗f〉 = A〈U∗K∗(U−1)∗f,U∗K∗(U−1)∗f〉

≤ A||U∗||2〈K∗(U−1)∗f,K∗(U−1)∗f〉

≤ ||U||2
∑
j∈J

v2j〈ΛjPWjU
∗PUWj (U

−1)∗f, ΛjPWjU
∗PUWj (U

−1)∗f〉

≤ ||U||2
∑
j∈J

v2j〈ΛjPWjU
∗(U−1)∗f, ΛjPWjU

∗(U−1)∗f〉

= ||U||2
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉.

Then, for each f ∈H, we have

A

||U||2
〈(U−1KU)∗f, (U−1KU)∗f〉≤

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉.



Int. J. Anal. Appl. 19 (6) (2021) 845

Also, for each f ∈H, we have

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉 =
∑
j∈J

v2j〈ΛjPWjU
∗(U−1)∗f, ΛjPWjU

∗(U−1)∗f〉

=
∑
j∈J

v2j〈ΛjPWjU
∗PUWj (U

−1)∗f, ΛjPWjU
∗PUWj (U

−1)∗f〉

≤ B〈(U−1)∗f, (U−1)∗f〉

≤ B||U−1||2〈f,f〉.

Thus, Λ is a U−1KU −g−fusion frame for H. �

Theorem 2.7. Let K ∈ End∗A(H) be an invertible operator on H and Λ = (Wj, Λj,vj)j∈J be a g−fusion

frame for H with frame bounds A, B and SΛ be the associated g−fusion frame operator. Suppose that for

all j ∈ J, T∗TWj ⊂ Wj, where T = KS−1Λ . Then (KS
−1
Λ Wj, ΛjPWjS

−1
Λ K

∗,vj)j∈J is a K −g−fusion frame

for H with the corresponding g−fusion frame operator KS−1Λ K
∗.

Proof. We now T = KS−1Λ is invertible on H and T
∗ = (KS−1Λ )

∗ = S−1Λ K
∗. For each f ∈H, we have

〈K∗f,K∗f〉 = 〈SΛS−1Λ K
∗f,SΛS−1Λ K

∗f〉

≤ ||SΛ||2〈S−1Λ K
∗f,S−1Λ K

∗f〉

≤ B2〈S−1Λ K
∗f,S−1Λ K

∗f〉.

Now for each f ∈H, we get

∑
j∈J

v2j〈ΛjPWjT
∗PT Wj (f), ΛjPWjT

∗PT Wj (f)〉 =
∑
j∈J

v2j〈ΛjPWjT
∗(f), ΛjPWjT

∗(f)〉

≤ B〈T∗f,T∗f〉

≤ B||T ||2〈f,f〉

≤ B||S−1Λ ||
2||K||2〈f,f〉

≤
B

A2
||K||2〈f,f〉.

On the other hand, for each f ∈H, we have

∑
j∈J

v2j〈ΛjPWjT
∗PT Wj (f), ΛjPWjT

∗PT Wj (f)〉 =
∑
j∈J

v2j〈ΛjPWjT
∗(f), ΛjPWjT

∗(f)〉

≥ A〈T∗f,T∗f〉

= A〈S−1Λ K
∗f,S−1Λ K

∗f〉

≥
A

B2
〈K∗f,K∗f〉.



Int. J. Anal. Appl. 19 (6) (2021) 846

Thus (KS−1Λ Wj, ΛjPWjS
−1
Λ K

∗,vj)j∈J is a K −g−fusion frame for H.

For each f ∈H, we have

∑
j∈J

v2j PT Wj (ΛjPWjT
∗)∗(ΛjPWjT

∗)PT Wjf =
∑
j∈J

v2j PT WjTPWj Λ
∗
j (ΛjPWjT

∗)PT Wjf

=
∑
j∈J

v2j (PWjT
∗PT Wj )

∗Λ∗j Λj(PWjT
∗PT W j)f

=
∑
j∈J

v2j TPWj Λ
∗
j ΛjPWjT

∗f

= T(
∑
j∈J

v2j PWj Λ
∗
j ΛjPWjT

∗f)

= TSΛT∗(f) = KS−1Λ K
∗(f).

This implies that KS−1Λ K
∗ is the associated g−fusion frame operator. �

Theorem 2.8. Let Λ = (Wj, Λj,vj)j∈J be a K − g−fusion frame for H with bounds A, B and for each

j ∈ J, Tj ∈ End∗A(Kj) be invertible operator. Suppose

0 < m = inf
j∈J

1

||T−1j ||
≤ sup

j∈J
||Tj|| = M.

If T ∈ End∗A(H) is an invertible operator on H with KT = TK and T
∗TWj ⊂ Wj, ∀j ∈ J then Γ =

(TWj,TjΛjPWjT
∗,vj)j∈J is a K −g−fusion frame for H.

Proof. Since T and Tj(for each j ∈ J) are invertible, so

〈K∗f,K∗f〉 = 〈(T−1)∗T∗K∗f, (T−1)∗T∗K∗f〉

≤ ||(T−1)||2〈T∗K∗f,T∗K∗f〉,

〈f,f〉 = 〈T−1j Tjf,T
−1
j Tjf〉

≤ ||T−1j ||
2〈Tjf,Tjf〉.

For each f ∈H, we have

∑
j∈J

v2j〈TjΛjPWjT
∗PT Wjf,TjΛjPWjT

∗PT Wjf〉 =
∑
j∈J

v2j〈TjΛjPWjT
∗f,TjΛjPWjT

∗f〉

≤
∑
j∈J
||Tj||2v2j〈ΛjPWjT

∗f, ΛjPWjT
∗f〉

≤ M2
∑
j∈J

v2j〈ΛjPWjT
∗f, ΛjPWjT

∗f〉

≤ M2B〈T∗f,T∗f〉

≤ M2B||T ||2〈f,f〉.



Int. J. Anal. Appl. 19 (6) (2021) 847

On the other hand, for each f ∈H, we have

∑
j∈J

v2j〈TjΛjPWjT
∗PT Wjf,TjΛjPWjT

∗PT Wjf〉 =
∑
j∈J

v2j〈TjΛjPWjT
∗f,TjΛjPWjT

∗f〉

≥
∑
j∈J

1

||T−1j ||2
v2j〈ΛjPWjT

∗f, ΛjPWjT
∗f〉

≥ m2
∑
j∈J

v2j〈ΛjPWjT
∗f, ΛjPWjT

∗f〉

≥ m2A〈K∗T∗f,K∗T∗f〉

≥
m2A

||(T−1)||2
〈K∗f,K∗f〉.

Thus, Γ is a K −g−fusion frame for H. �

In this theorem we give a necessary and sufficient condition for a quotient operator to be bounded.

Theorem 2.9. Let K ∈ End∗A(H), and Λ = (Wj, Λj,vj)j∈J be a K − g−fusion frame for H with frame

operator SΛ and frame bounds A and B. Let U ∈ End∗A(H) be an invertible operator on H, and U
∗UWj ⊂

Wj,∀j ∈ J. Then the following statements are equivalent:

(1) Γ = (UWj, ΛjPWjU
∗,vj)j∈J is a UK −g−fusion frame.

(2) The quotient operator [(UK)∗/S
1
2

Λ U
∗] is bounded.

(3) The quotient operator [(UK)∗/(USΛU∗)
1
2 ] is bounded.

Proof. (1) =⇒ (2) Since Γ is K −g−fusion frame then there exist A, B > 0 such that for each f ∈H

A〈(UK)∗f, (UK)∗〉≤
∑
j∈J

v2j〈ΛjPWjU
∗PUWjf, ΛjPWjU

∗PUWjf〉≤ B〈f,f〉

For each f ∈H we have

∑
j∈J

v2j〈ΛjPWjU
∗PUWjf, ΛjPWjU

∗PUWjf〉 =
∑
j∈J

v2j〈ΛjPWjU
∗f, ΛjPWjU

∗f〉

= 〈SΛU∗f,U∗f〉

= 〈S
1
2

Λ U
∗f,S

1
2

Λ U
∗f〉.

Then

A〈(UK)∗f, (UK)∗f〉≤ 〈S
1
2

Λ (U
∗f),S

1
2

Λ (U
∗f)〉.

We define the operator: T : R(S
1
2

Λ U
∗) →R((UK)∗) by

T(S
1
2

Λ U
∗f) = (UK)∗f, ∀f ∈H.



Int. J. Anal. Appl. 19 (6) (2021) 848

T is linear operator and Ker(S
1
2

Λ U
∗) ⊂ Ker((UK)∗). Thus T is well-defined quotient operator. Therefore

for each f ∈H

||T(S
1
2

Λ U
∗f)|| = ||(UK)∗f||

≤
1
√
A
||〈S

1
2

Λ U
∗f,S

1
2

Λ U
∗f〉||

1
2

≤
1
√
A
||S

1
2

Λ U
∗f||.

So T is bounded.

(2) =⇒ (3) Suppose that the quotion operator [(UK)∗/S
1
2

Λ U
∗] is bounded. Thus for all f ∈H, ∃C > 0

such that

||(UK)∗f|| ≤ C||S
1
2

Λ U
∗f||

≤ C||〈S
1
2

Λ U
∗f,S

1
2

Λ U
∗f〉||

1
2

≤ C||〈USΛU∗f,f〉||
1
2

≤ C||〈(UKU∗)
1
2 f, (UKU∗)

1
2 f〉||

1
2

≤ C||(USΛU∗)
1
2 f||.

Hence the quotient operator [(UK)∗/(USΛU∗)
1
2 ] is bounded.

(3) =⇒ (1) For each f ∈H, we have

∑
j∈J

v2j〈ΛjPWjU
∗PUWjf, ΛjPWjU

∗PUWjf〉 =
∑
j∈J

v2j〈ΛjPWjU
∗f, ΛjPWjU

∗f〉

≥ A〈K∗(U∗f),K∗(U∗f)〉

= A〈(UK)∗f, (UK)∗f〉.

On the other hand for each f ∈H

∑
j∈J

v2j〈ΛjPWjU
∗PUWjf, ΛjPWjU

∗PUWjf〉 =
∑
j∈J

v2j〈ΛjPWjU
∗f, ΛjPWjU

∗f〉

≤ B〈U∗f,U∗f〉

≤ B||U||2〈f,f〉.

Hence Γ is a UK −g−fusion frame for H. �

Studying g−fusion frame in Hilbert C∗-modules with different C∗-algebras is interesting and important.

In the following, we study this situation.

In the next theorem we take Kj ⊂H for each j ∈ J.



Int. J. Anal. Appl. 19 (6) (2021) 849

Theorem 2.10. Let (H,A,〈., .〉A) and (H,B,〈., .〉B) be two Hilbert C∗−modules and φ : A → B be a

∗−homomorphism and θ be a map on H such that 〈θf,θg〉B = φ(〈f,g〉A) for all f,g ∈H. Also, suppose that

Λ = (Wj, Λj,vj)j∈J is a g−fusion frame for (H,A,〈., .〉A) with g−fusion frame operator SA and lower and

upper g−fusion frame bounds A, B respectively. If θ is surjective and θΛjPWj = ΛjPWjθ for each j ∈ J, then

Λ = (Wj, Λj,φ(vj))j∈J is a g−fusion frame for (H,B,〈., .〉B) with g−fusion frame operator SB and lower

and upper g−fusion frame bounds φ(A), φ(B) respectively, and 〈SBθf,θg〉B = φ(〈SAf,g〉A).

Proof. Let g ∈H then there exists f ∈H such that θf = g. By the definition of g−fusion frame we have

A〈f,f〉A ≤
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉A ≤ B〈f,f〉A.

Then

φ(A〈f,f〉A) ≤
∑
j∈J

φ(v2j〈ΛjPWjf, ΛjPWjf〉A) ≤ φ(B〈f,f〉A).

By the definition of ∗−homomorphism we have

Aφ(〈f,f〉A) ≤
∑
j∈J

φ(v2j )φ(〈ΛjPWjf, ΛjPWjf〉A) ≤ Bφ(〈f,f〉A).

By the relation betwen θ and φ we get

A〈θf,θf〉B ≤
∑
j∈J

φ(vj)
2〈θΛjPWjf,θΛjPWjf〉B ≤ B〈θf,θf〉B.

Then

A〈θf,θf〉B ≤
∑
j∈J

φ(vj)
2〈ΛjPWjθf, ΛjPWjθf〉B ≤ B〈θf,θf〉B.

So, we have

A〈g,g〉B ≤
∑
j∈J

φ(vj)
2〈ΛjPWjg, ΛjPWjg〉B ≤ B〈g,g〉B, ∀g ∈H.



Int. J. Anal. Appl. 19 (6) (2021) 850

On the other hand we have

φ(〈SAf,g〉A) = φ(〈
∑
j∈J

v2j PWj Λ
∗
j ΛjPWjf,g〉A)

=
∑
j∈J

φ(v2j〈ΛjPWjf, ΛjPWjg〉A)

=
∑
j∈J

φ(vj)
2〈θΛjPWjf,θΛjPWjg〉B

=
∑
j∈J

φ(vj)
2〈ΛjPWjθf, ΛjPWjθg〉B

=
∑
j∈J

φ(vj)
2〈PWj Λ

∗
j ΛjPWjθf,θg〉B

= 〈
∑
j∈J

φ(vj)
2PWj Λ

∗
j ΛjPWjPWjθf,θg〉B

= 〈SBθf,θg〉B.

�

3. stability of G-fusion frames in hilbert C∗−modules

Frome Theorem 2.7 if Λ = (Wj, Λj,vj)j∈J is a g−fusion frame for H with associated frame operator SΛ,

such that S−2Λ Wj ⊂ Wj, for all j ∈ J then Λ̃ = (S
−1
Λ Wj, ΛjPWjS

−1
Λ ,vj)j∈J is called the canonical dual

g−fusion frame of Λ. The frame operator SΛ̃ of Λ̃ is described by, for each f ∈H

SΛ̃(f) =
∑
j∈J

v2j PS−1
Λ

Wj
(ΛjPWjS

−1
Λ )
∗(ΛjPWjS

−1
Λ )PS−1

Λ
Wj

(f)

=
∑
j∈J

v2j PS−1
Λ

Wj
S−1Λ PWj Λ

∗
j Λj(PWjS

−1
Λ PS−1

Λ
Wj

)(f)

=
∑
j∈J

v2j (PWjS
−1
Λ PS−1

Λ
Wj

)∗Λ∗j Λj(PWjS
−1
Λ PS−1

Λ
Wj

)(f)

=
∑
j∈J

v2j (PWjS
−1
Λ )
∗Λ∗j ΛjPWjS

−1
Λ (f)

=
∑
j∈J

v2jS
−1
Λ PWj Λ

∗
j ΛjPWjS

−1
Λ (f)

= S−1Λ
∑
j∈J

v2j PWj Λ
∗
j ΛjPWjS

−1
Λ (f)

= S−1Λ (SΛ(S
−1
Λ f)) = S

−1
Λ (f).

Theorem 3.1. Let Λ = (Wj, Λj,vj)j∈J and Γ = (Vj, Γj,vj)j∈J be two g−fusion frames for H with lower

frame bounds A and C, respectively. Suppose that S−2Λ Wj ⊂ Wj and S
−2
Γ Vj ⊂ Vj, ∀j ∈ J. If there exist real



Int. J. Anal. Appl. 19 (6) (2021) 851

constant D > 0 such that for all f ∈H

||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉−
∑
j∈J

v2j〈ΓjPVjf, ΓjPVjf〉|| ≤ D||〈f,f〉||.

Then for all f ∈H

||
∑
j∈J

v2j〈Λ̃jPW̃j (f), Λ̃jPW̃j (f)〉−
∑
j∈J

v2j〈Γ̃jPṼj (f), Γ̃jPṼj (f)〉|| ≤
D

AC
||〈f,f〉||.

Such that W̃j = S−1Λ Wj, Λ̃j = ΛjPWjS
−1
Λ , Ṽj = S

−1
Γ Vj, Γ̃j = ΓjPVjS

−1
Γ .

Proof. We have for every f ∈H

||SΛ −SΓ|| = sup
||f||=1

||〈(SΛ −SΓ)f,f〉||

= sup
||f||=1

||〈SΛf,f〉−〈SΓf,f〉||

= sup
||f||=1

||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉−
∑
j∈J

v2j〈ΓjPVjf, ΓjPVjf〉|| ≤ D.

Therefore,

||S−1Λ −S
−1
Γ || ≤ ||S

−1
Λ ||||SΛ −SΓ||||S

−1
Γ || ≤

D

AC
.

And for all f ∈H

∑
j∈J

v2j〈ΛjPWjS
−1
Λ PS−1

Λ
Wj
f, ΛjPWjS

−1
Λ PS−1

Λ
Wj
f〉 =

∑
j∈J

v2j〈ΛjPWjS
−1
Λ f, ΛjPWjS

−1
Λ f〉

=
∑
j∈J

v2j〈PWj Λ
∗
j ΛjPWjS

−1
Λ f,S

−1
Λ f〉

= 〈
∑
j∈J

v2j PWj Λ
∗
j ΛjPWj (S

−1
Λ f),S

−1
Λ f〉

= 〈SΛ(S−1Λ ),S
−1f〉

= 〈f,S−1Λ f〉.

Similarly we have for all f ∈H

∑
j∈J

v2j〈ΓjPVjS
−1
Γ PS−1

Γ
Vj
f, ΓjPVjS

−1
Γ PS−1

Γ
Vj
f〉 = 〈f,S−1Γ f〉.

Then

||
∑
j∈J

v2j〈ΛjPWjS
−1
Λ PS−1

Λ
Wj
f, ΛjPWjS

−1
Λ PS−1

Λ
Wj
f〉−

∑
j∈J
〈ΓjPVjS

−1
Γ PS−1

Γ
Vj
f, ΓjPVjS

−1
Γ PS−1

Γ
Vj
f〉||



Int. J. Anal. Appl. 19 (6) (2021) 852

= ||〈f,S−1Λ f〉−〈f,S
−1
Γ f〉||

= ||〈f, (S−1Λ −S
−1
Γ )f〉||

≤ ||S−1Λ −S
−1
Γ ||||f||

2

≤
D

AC
||〈f,f〉||.

�

Now we give a characterazation of g−fusion frames for Hilbert A−modules.

Theorem 3.2. Let H be a Hilbert A−module over C∗−algebra. Then Λ = (Wj, Λj,vj)j∈J is a g−fusion

frame for H if and only if there exist two constants 0 < A ≤ B < ∞ such that for all f ∈H

A||〈f,f〉|| ≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉|| ≤ B||〈f,f〉||.

Proof. Suppose Λ is g−fusion frame for H, since there is 〈f,f〉≥ 0 then for all f ∈H,

A||〈f,f〉|| ≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉|| ≤ B||〈f,f〉||

Conversely for each f ∈H we have

||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉|| = ||
∑
j∈J
〈vjΛjPWjf,vjΛjPWjf〉||

= ||〈(vjΛjPWjf)j∈J, (vjΛjPWjf)j∈J〉||

= ||(vjΛjPWjf)j∈J||
2.

We define the operator L : H→ l2((Kj)j∈J) by L(f) = (vjΛjPWjf)j∈J, then

||L(f)||2 = ||(vjΛjPWjf)j∈J||
2 ≤ B||f||2.

L is A−linear bounded operator, then there exist C > 0 sutch that

〈L(f),L(f)〉≤ C〈f,f〉, ∀f ∈H.

So ∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ C〈f,f〉, ∀f ∈H.

Therefore Λ = (Wj, Λj,vj)j∈J is g−fusion bessel sequence for H. Now we cant define the g−fusion frame

operator SΛ on H. So ∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉 = 〈SΛf,f〉, ∀f ∈H.

Since SΛ is self-adjoint and positive, then

〈S
1
2

Λ f,S
1
2

Λ f〉 = 〈SΛf,f〉, ∀f ∈H.



Int. J. Anal. Appl. 19 (6) (2021) 853

That implies

A||〈f,f〉|| ≤ ||〈S
1
2

Λ f,S
1
2

Λ f〉|| ≤ B||〈f,f〉||, ∀f ∈H.

Frome lemma1.2 there exist two canstants A
′
,B
′
> 0 such that

A
′
〈f,f〉≤ 〈S

1
2

Λ f,S
1
2

Λ f〉≤ B
′
〈f,f〉, ∀f ∈H.

So

A
′
〈f,f〉≤

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ B
′
〈f,f〉, ∀f ∈H.

Hence Λ is a g−fusion frame for H. �

Theorem 3.3. Let Λ = (Wj, Λj,vj)j∈J be a g−fusion frame for H with frame bounds A and B. If Γ =

(Wj, Γj,vj)j∈J is g−fusion bessel sequence with bound M < A, then (Wj, Λj + Γj,vj)j∈J is g−fusion frame

for H.

Proof. Let f ∈H, we have

||
∑
j∈J

v2j〈(Λj + Γj)PWjf, (Λj + Γj)PWjf〉||
1
2 = ||(vj(Λj + Γj)PWjf)j∈J||.

≤ ||(vjΛjPWjf)j∈J|| + ||(vjΓjPWjf)j∈J||

≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 + ||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2

≤
√
B||f|| +

√
M||f||

≤ (
√
B +

√
M)||f||.

On the other hand, for each f ∈H

||
∑
j∈J

v2j〈(Λj + Γj)PWjf, (Λj + Γj)PWjf〉||
1
2 = ||(vj(Λj + Γj)PWjf)j∈J||.

≥ ||(vjΛjPWjf)j∈J||− ||(vjΓjPWjf)j∈J||

≥ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 −||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2

≥
√
A||f||−

√
M||f||

≥ (
√
A−
√
M)||f||.

So,

(
√
A−
√
M)2||f||2 ≤ ||

∑
j∈J

v2j〈(Λj + Γj)PWjf, (Λj + Γj)PWjf〉|| ≤ (
√
B +

√
M)2||f||2, ∀f ∈H.

Hence (Wj, Λj + Γj,vj)j∈J is g−fusion frame for H. �



Int. J. Anal. Appl. 19 (6) (2021) 854

Theorem 3.4. Let (Wj, Λj,vj)j∈J be a g−fusion frame for H with frame bounds A and B. And Γj ∈

End∗A(H,Kj), ∀j ∈ J. Then the following statements are equivalent:

(1) (Wj, Γj,vj)j∈J is g−fusion frame for H.

(2) There exist a canstant M such that ∀f ∈H we have:

||
∑
j∈J

v2j〈(Λj − Γj)PWjf, (Λj − Γj)PWjf〉|| ≤ M min(||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||,

||
∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||).

Proof. (1) =⇒ (2) Let (Wj, Γj,vj)j∈J be a g−fusion frame for H, with frame bounds C and D, then for

any f ∈H, we have

||
∑
j∈J

v2j〈(Λj − Γj)PWjf,(Λj − Γj)PWjf〉||
1
2 = ||(vj(Λj − Γj)PWjf)j∈J||

≤ ||(vjΛjPWjf)j∈J|| + ||(vjΓjPWjf)j∈J||

≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 + ||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2

≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 +
√
D||f||

≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 +

√
D
√
A
||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2

≤ (1 +
√
D

A
)||

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 .

Similary, for each f ∈H, we can obtain

||
∑
j∈J

v2j〈(Λj − Γj)PWjf, (Λj − Γj)PWjf〉||
1
2 ≤ (1 +

√
B

C
)||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2 .

We put M = min((1 +
√

B
C

)2, (1 +
√

D
A

)2).

(2) =⇒ (1) We have for each f ∈H

√
A||f|| ≤ ||

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2

≤ ||(vjΛjPWjf)j∈J||

≤ ||(vj(Λj − Γj)PWjf)j∈J|| + ||(vjΓjPWjf)j∈J||

≤
√
M||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2 + ||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2

≤ (
√
M + 1)||

∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2 .



Int. J. Anal. Appl. 19 (6) (2021) 855

And we have ∀f ∈H

||
∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉||
1
2 = ||(vjΓjPWjf)j∈J||

≤ ||(vjΛjPWjf)j∈J|| + ||(vj(Λj − Γj)f)j∈J||

≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2 +
√
M||

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2

≤ (
√
M + 1)||

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉||
1
2

≤ (
√
M + 1)

√
B||f||.

So
A

(1 +
√
M)2

||f||2 ≤ ||
∑
j∈J

v2j〈ΓjPWjf, ΓjPWjf〉|| ≤ (
√
M + 1)2B||f||2, ∀f ∈H.

Hence (Wj, Γj,vj)j∈J is g−fusion frame for H. �

4. perturbation of K-g-fusion frames

Perturbation of frames has been discussed by Casazza and Christensen. In this section, we present a

perturbation of K − g−fusion frames. first we give a characterazation of K − g−fusion frame for Hilbert

A−modules.

Theorem 4.1. Let K ∈ End∗A(H). Suppose that the operator T : H → l
2((Kj)j∈J) is given by T(f) =

(vjΛjPWjf)j∈J, ∀f ∈H, and R(T) is orthogonally complemented. Then Λ = (Wj, Λj,vj)j∈J is K−g−fusion

frame for H if and only if there exist constants 0 < A ≤ B < ∞ such that

(4.1) A||K∗f||2 ≤ ||
∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉|| ≤ B||f||
2, ∀f ∈H.

Proof. If Λ is K − g−fusion frame for H, then the equation (4.1) is satisfies. Conversly, we have for each

(fj)j∈J ∈ l2((Kj)j∈J) and any finite I ⊂ J

||
∑
j∈I

vjPWj Λ
∗
j (fj)|| = sup

||g||=1
||〈
∑
j∈I

vjPWj Λ
∗
jfj,g〉||

= sup
||g||=1

||
∑
j∈I
〈fj,vjΛjPWjg〉||

≤ sup
||g||=1

||
∑
j∈I
〈fj,fj〉||

1
2 ||

∑
j∈I

v2j〈ΛjPWjg〉||
1
2

≤
√
B||

∑
j∈I
〈fj,fj〉||

1
2 .

Then
∑

j∈J vjPWj Λ
∗
jfj converge unconditionally in H, and we have ∀f ∈H, ∀(fj)j∈J ∈ l

2((Kj)j∈J)

〈Tf, (fj)j∈J〉 = 〈(vjΛjPWjf)j∈J, (fj)j∈J〉 =
∑
j∈J
〈vjΛjPWjf,fj〉 = 〈f,

∑
j∈J

vjPWj Λ
∗
jfj〉



Int. J. Anal. Appl. 19 (6) (2021) 856

So T is adjointable and T∗((fj)j∈J) =
∑

j∈J vjPWj Λ
∗
jfj also frome (4.1) we have

||K∗f||2 ≤
1

A
||Tf||2, ∀f ∈H.

By lemma1.5, there exist ν > 0 such that KK∗ ≤ νT∗T , then

〈KK∗f,f〉≤ ν〈Tf,Tf〉, ∀f ∈H.

Therefore

1

ν
〈K∗f,K∗f〉≤

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉, ∀f ∈H.

And we have for each f ∈H, 〈Tf,Tf〉≤ ||T ||2〈f,f〉, then

∑
j∈J

v2j〈ΛjPWjf, ΛjPWjf〉≤ ||T ||
2〈f,f〉, ∀f ∈H.

And the proof is completed. �

Theorem 4.2. Let Λ = (Wj, Λj,vj)j∈J be a K − g−fusion frame for H with frame bounds A, B and let

Γj ∈ End∗A(H,Kj), for all j ∈ J. Suppose that T : H → l
2((Kj)j∈J) define by T(f) = (ujΓjPVjf)j∈J,

∀f ∈H. And R(T) is orthogonally complemented, such that for each f ∈H

||((vjΛjPWj −ujΓjPVj )f)j∈J|| ≤ λ1||(vjΛjPWjf)j∈J|| + λ2||(ujΓjPVjf)j∈J|| + �||K
∗f||.

where 0 < λ1,λ2 < 1 and � > 0 such that � < (1 −λ1)
√
A.

Then Γ = (Vj, Γj,uj)j∈J is a K −g−fusion frame for H.

Proof. We have for each f ∈H

||
∑
j∈J

u2j〈ΓjPVjf, ΓjPVjf〉||
1
2 = ||(ujΓjPVjf)j||

= ||(ujΓjPVjf)j + (vjΛjPWjf)j − (vjΛjPWjf)j||

≤ ||((ujΓjPVj −vjΛjPWj )f)j|| + ||(vjΛjPWjf)j||

≤ (λ1 + 1)||(vjΛjPWjf)j|| + λ2||(ujΓjPVjf)j|| + �||K
∗f||.

So

(1 −λ2)||(ujΓjPVjf)j|| ≤ (λ1 + 1)
√
B||f|| + �||K∗f||.

Then

||(ujΓjPVjf)j|| ≤
(λ1 + 1)

√
B||f|| + �||K∗f||
1 −λ2

≤ (
(λ1 + 1)

√
B + �||K||

1 −λ2
)||f||.



Int. J. Anal. Appl. 19 (6) (2021) 857

Hence

||
∑
j∈J

u2j〈ΓjPVjf, ΓjPVjf〉|| ≤ (
(λ1 + 1)

√
B + �||K∗||

1 −λ2
)2||f||2.

On the other hand for each f ∈H

||
∑
j∈J

u2j〈ΓjPVjf, ΓjPVjf〉||
1
2 = ||(ujΓjPVjf)j||

= ||((ujΓjPVj −vjΛjPWj )f)j + (vjΛjPWjf)j||

≥ ||(vjΛjPWjf)j||− ||((ujΓjPVj −vjΛjPWj )f)j||

≥ (1 −λ1)||(vjΛjPWjf)j||−λ2||(ujΓjPVjf)j||− �||K
∗f||.

Hence

||
∑
j∈J

u2j〈ΓjPVjf, ΓjPVjf〉|| ≥ (
(1 −λ1)

√
A− �

1 + λ2
)2||K∗f||2.

�

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction and Preliminaries
	2. K-g-fusion frames in Hilbert C-modules 
	3. stability of G-fusion frames in hilbert C-modules
	4. perturbation of K-g-fusion frames
	References