©2023 Ada Academica https://adac.eeEur. J. Math. Anal. 3 (2023) 11doi: 10.28924/ada/ma.3.11
Woven K − g−Fusion Frames in Hilbert C∗−Modules

Fakhr-dine Nhari1, Mohamed Rossafi2,∗

1Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty Of Sciences,
University of Ibn Tofail, P. O. Box 133 Kenitra, Morocco

nharidoc@gmail.com
2LaSMA Laboratory, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi

Mohamed Ben Abdellah, P. O. Box 1796 Fez Atlas, Morocco
rossafimohamed@gmail.com

∗Correspondence: rossafimohamed@gmail.com

Abstract. In this paper, we introduced the notion of woven K − g−fusion frames in Hilbert
C∗−modules. We present necessary and sufficient conditions for these woven and also constructthem by linear bounded operator. Finally we study perturbation of weaving K − g−fusion frames.

1. Introduction
Basis is one of the most important concepts in Vector Spaces study. However, Frames generaliseorthonormal bases and were introduced by Duffin and Schaefer [3] in 1952 to analyse some deepproblems in nonharmonic Fourier series by abstracting the fundamental notion of Gabor [5] for signalprocessing. In 2000, Frank-larson [4] introduced the concept of frames in Hilbet C∗−modulesas a generalization of frames in Hilbert spaces. The basic idea was to consider modules over

C∗−algebras of linear spaces and to allow the inner product to take values in the C∗−algebras [6].Many generalizations of the concept of frame have been defined in Hilbert C∗-modules [7,9,11–16].Throughout this paper, H is considered to be a countably generated Hilbert C∗−module. Let
{Hj}j∈J are the collection of Hilbert C∗−module and {Wj}j∈J is a collection of closed orthogonallycomplemented submodules of H, where J be finite or countable index set. End∗A(H,Hj) is a setof all adjointable operator from H to Hj . In particular End∗A(H) denote the set of all adjointableoperators on H. PWj

denote the orthogonal projection onto the closed submodule orthogonally
Received: 31 Jul 2022.2010 Mathematics Subject Classification. Primary 41A58; Secondary 42C15.
Key words and phrases. fusion frames; K − g−fusion frames; woven K − g−fusion frames; C∗-algebra; Hilbert

C∗-modules. 1

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https://doi.org/10.28924/ada/ma.3.11


Eur. J. Math. Anal. 10.28924/ada/ma.3.11 2complemented Wj of H. Define the module
l2({Hj}j∈J) = {{fj}j∈J : fj ∈ Hj , ‖

∑
j∈J
〈fj , fj〉‖ <∞}

with A−valued inner product 〈f , g〉 =
∑
j∈J〈fj , gj〉, where f = {fj}j∈J and g = {gj}j∈J, clearly

l2({Hj}j∈J) is a Hilbert A−module.
Definition 1.1. [8] Let A be a unital C∗-algebra and H be a left A-module, such that the linearstructures 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 respectsthe 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-moduleor 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.

Lemma 1.2. [10] 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 complemented closed submodules and PWj

T ∗ = PWj
T ∗PTWj

.

Lemma 1.3. [2]. Let H and K two Hilbert A-modules and T ∈ End∗A(H,K). Then the following
statements are equivalent:(i) T is surjective.(ii) T ∗ is bounded below with respect to norm, i.e., there is m > 0 such that ‖T ∗x‖ ≥ m‖x‖

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

Lemma 1.4. [1]. Let U and H two Hilbert A-modules and T ∈ End∗A(U,H). 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.

Definition 1.5. [10] Let {Wi}i∈I be a sequence of closed orthogonally complemented submodulesof H, {vi}i∈I be a familly of positive weights in A, i.e., each vi is a positive invertible element from

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Eur. J. Math. Anal. 10.28924/ada/ma.3.11 3the center of the C∗−algebra A and Λi ∈ End∗A(H,Hi) for all i ∈ I . We say that Λ = {Wi ,Λi , vi}i∈Iis a g−fusion frame for H if and only if there exists two constants 0 < A ≤ B <∞ such that
A〈x, x〉 ≤

∑
i∈I

v2
i 〈ΛiPWi

x,ΛiPWi
x〉 ≤ B〈x, x〉, ∀x ∈ H. (1.1)

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−fusionframe. If Λ satisfies the inequality∑

i∈I
v2
i 〈ΛiPWi

x,ΛiPWi
x〉 ≤ B〈x, x〉, ∀x ∈ H.

then it is called a g−fusion bessel sequence with bound B in H.
Definition 1.6. [10]let Λ = {Wj ,Λj , vj}j∈J be a g−fusion bessel sequence for H. Then the operator TΛ :

l2({Hj}j∈J)→ H defined by
TΛ({fj}j∈J) =

∑
j∈J

vjPWj
Λ∗j fj , ∀{fj}j∈J ∈ l2({Hj}j∈J).

Is called synthesis operator. We say the adjoint UΛ of the synthesis operator the analysis operatorand it is defined by UΛ : H → l2({Hj}j∈J) such that
UΛ(f ) = {vjΛjPWj

(f )}j∈J, ∀f ∈ H.

The operator SΛ : H → H defined by
SΛf = TΛUΛf =

∑
j∈J

v2
j PWj

Λ∗j ΛjPWj
(f ), ∀f ∈ H.

Is called g−fusion frame operator. It can be easily verify that
〈SΛf , f 〉 =

∑
j∈J

v2
j 〈ΛjPWj

(f ),ΛjPWj
(f )〉, ∀f ∈ H. (1.2)

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 inversibilityof SΛ. Let f ∈ H we have
||UΛ(f )|| = ||{vjΛjPWj

(f )}j∈I || = ||
∑
j∈J

v2
j 〈ΛjPWj

(f ),ΛjPWj
(f )〉||

1
2 .

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

1
2 ≤ ||UΛf ||.Then

√
A||f || ≤ ||UΛf ||.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.11 4Frome lemma 1.3, TΛ is surjective and by lemma 1.4, TΛUΛ = SΛ is invertible. We now, AIH ≤
SΛ ≤ BIH and this gives B−1IH ≤ S−1

Λ ≤ A−1IH .
2. Woven K − g−fusion frames in Hilbert C∗−modules

Throughout this paper, [m] = {1, 2, ..., m} for each m > 1, {Wi j}j∈J,i∈[m] is a collection of closedorthogonally complemented submodules of H, {vi j}j∈J,i∈[m] is a family of weights, K ∈ End∗A(H)and {Λi j}j∈J,i∈[m] ∈ End∗A(H,Hi j) where Hi j are Hilbert A−modules.
Definition 2.1. A family of g−fusion frames {Wi j ,Λi j , vi j}j∈J,i∈[m] for H is said to be K− g−fusionwoven if there exist universal positive constants 0 < A ≤ B such that for each partition {σi}i∈[m]of J, the family {Wi j ,Λi j , vi j}j∈σi ,i∈[m] is a K − g−fusion frame for H with bounds A and B.

In next theorem, we provide a necessary and sufficient condition for weaving K−g−fusion frames.
Theorem 2.2. Assume that {Wj ,Λj , vj}j∈J and {Vj , θj , µj}j∈J are two K − g−fusion frames for
H where Λj ∈ End∗A(H,Hj) and θj ∈ End∗A(H,Hj) for any j ∈ J, the following assertions are
equivalent.(1) {Wj ,Λj , vj}j∈J and {Vj , θj , µj}j∈J are K − g−fusion woven.(2) there exists α > 0 such that for each σ ⊂ J there exists a bounded linear operator

ψσ : lσ2 ({Hj}j∈J)→ H,

ψσ{xj}j∈J =
∑
j∈σ

vjPWj
Λ∗j xj +

∑
j∈σc

µjPVj θ
∗
j xj ,

such that αKK∗ ≤ ψσψ∗σ , where

lσ2 ({Hj}j∈J) = {{xj}j∈J = {fj}j∈σ ∪ {gj}j∈σc : fj ∈ Hj , gj ∈ Hj , ‖
∑
j∈J
〈xj , xj〉‖ <∞}.

Proof. (1) =⇒ (2): Suppose that A is an universal lower frame bound for {Wj ,Λj , vj}j∈J and
{Vj , θj , µj}j∈J. Choose α = A and ψσ = Tσ for every σ ⊂ J, where Tσ is the synthesis operator of
{Wj ,Λj , vj}j∈σ ∪ {Vj , θj , µj}j∈σc . Then, for any {xj}j∈J ∈ lσ2 ({Hj}j∈J) we have

ψσ{xj}j∈J = Tσ{xj}j∈J

=
∑
j∈σ

vjPWj
Λ∗j xj +

∑
j∈σc

µjPVj θ
∗
j xj ,

and also, for each f ∈ H,
A〈K∗f , K∗f 〉 ≤ 〈T ∗σ f , T ∗σ f 〉 = 〈ψ∗σf , ψ∗σf 〉.Thus, αKK∗ ≤ ψσψ∗σ .

(2) =⇒ (1): Let σ ⊂ J and f ∈ H, so it is easy to check that
ψ∗σf = {vjΛjPWj

f }j∈σ ∪ {µjθjPVj f }j∈σc .

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Eur. J. Math. Anal. 10.28924/ada/ma.3.11 5Therefore,
α〈K∗f , K∗f 〉 = 〈αKK∗f , f 〉

≤ 〈ψσψ∗σf , f 〉

= 〈ψ∗σf , ψ∗σf 〉

=
∑
j∈σ

v2
j 〈ΛjPWj

f ,ΛjPWj
f 〉+

∑
j∈σc

µ2
j 〈θjPVj f , θjPVj f 〉.

This gives that α is an universal lower frame bound of {Wj ,Λj , vj}j∈J and {Vj , θj , µj}j∈J. �

In next results, we construct a K − g−fusion woven by using a bounded linear operator.
Theorem 2.3. Let {Wi j ,Λi j , vi j}j∈J,i∈[m] be a K−g−fusion woven for H with common frame bounds
A,B and assume that U ∈ End∗A(H) has closed range so that R(K∗) ⊂ R(U) and KU = UK.
Then {UWi j ,Λi jPWi j

U∗, vi j}j∈J,i∈[m] is also K − g−fusion woven for R(U).

Proof. By the open mapping theorem, UWi j is closed for any j ∈ J and i ∈ [m]. Using Lemme(refk-g-fusion ), we can write for each f ∈ R(U),
A〈K∗f , K∗f 〉 = A〈(U+)∗U∗K∗f , (U+)∗U∗K∗f 〉

≤ A‖U+‖2〈K∗U∗f , K∗U∗f 〉

≤ ‖U+‖2
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

U∗f ,Λi jPWi j
U∗f 〉

= ‖U+‖2
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

U∗PUWi j
f ,Λi jPWi j

U∗PUWi j
f 〉.

The upper bound is obvious. �

Theorem 2.4. Let K have closed range, {Wi j ,Λi j , vi j}j∈J,i∈[m] be a K− g−fusion woven for H with
the universal bounds A,B and U ∈ End∗A(H) has closed range so that R(U∗) ⊂ R(K). Then
{UWi j ,Λi jPWi j

U∗, vi j}j∈J,i∈[m] is a K − g−fusion woven for H if and only if there exists a δ > 0

such that for every f ∈ H,

〈U∗f , U∗f 〉 ≥ δ〈K∗f , K∗f 〉.

Proof. Let f ∈ H and {UWi j ,Λi jPWi j
U∗, vi j}j∈J,i∈[m] is a K − g−fusion woven for H with lowerbound C, we get

C〈K∗f , K∗f 〉 ≤
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

U∗PUWi j
f ,Λi jPWi j

U∗PUWi j
f 〉

=
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

U∗f ,Λi jPWi j
U∗f 〉

≤ B〈U∗f , U∗f 〉.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.11 6

Therefore, 〈U∗f , U∗f 〉 ≥√C
B 〈K

∗f , K∗f 〉. For the opposite implication, we can write for all f ∈ H,
〈U∗f , U∗f 〉 = 〈(K+)∗K∗U∗f , (K+)∗K∗U∗f 〉 ≤ ‖K+‖2〈K∗U∗f , K∗U∗f 〉.

Hence, we have
Aδ‖K+‖−2〈K∗f , K∗f 〉 ≤ A‖K+‖−2〈U∗f , U∗f 〉

≤ A〈K∗U∗f , K∗U∗f 〉

≤
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

U∗f ,Λi jPWi j
U∗f 〉

=
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

U∗PUWi j
f ,Λi jPWi j

U∗PUWi j
f 〉

≤ B‖U‖2〈f , f 〉.

So, {UWi j ,Λi jPWi j
U∗, vi j}j∈J,i∈[m] is a K − g−fusion woven for H with frame bounds Aδ‖K+‖−2and B‖U‖2. �

Theorem 2.5. Let {Wi j ,Λi j , vi j}j∈J,i∈[m] be a K − g−fusion woven for H with common frame
bounds A and B. Suppose that 0 ≤ C ≤ |w (i)

j |
2 ≤ D < ∞ for any i ∈ [m] and j ∈ J, then

{Wi j , w
(i)
j Λi j , vi j}j∈J,i∈[m] is a K − g−fusion woven for H with frame bounds AC and BD.

Proof. For any partition {σi}i∈[m] of J and f ∈ H, we get
AC〈K∗f , K∗f 〉 = min

i∈[m]
|w (i)
j |

2A〈K∗f , K∗f 〉 ≤
∑
i∈[m]

∑
j∈σi

v2
i j 〈w

(i)
j Λi jPWi j

f , w
(i)
j Λi jPWi j

f 〉

≤ max
i∈[m]

|w (i)
j |

2B〈f , f 〉

= BD〈f , f 〉.

�

Theorem 2.6. Let I ⊂ J be arbitrary and {Wi j ,Λi j , vi j}j∈I,i∈[m] be a K − g−fusion woven for H.
Then {Wi j ,Λi j , vi j}j∈J,i∈[m] is a K − g−fusion woven.

Proof. Assume that σi ⊂ J, so σi ∩ I ⊂ I and A is the lower bound of {Wi j ,Λi j , vi j}j∈σi∩I,i∈[m], thenfor every f ∈ H we have
A〈K∗f , K∗f 〉 ≤

∑
i∈[m]

∑
j∈σi∩I

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉

≤
∑
i∈[m]

∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉.

This implies the statement. �

Next theorem is shows that even if one subspace is deleted, it dose not still remain a K−g−fusionwoven.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.11 7

Theorem 2.7. Let K has closed range, I ⊂ J and {Wi j ,Λi j , vi j}j∈J,i∈[m] be a K − g−fusion woven
for H with the bounds A,B. If

C =
∑
i∈[m]

∑
j∈I

v2
i j‖Λi jPWi j

‖2 < A‖K+‖2,

then {Wi j ,Λi j , vi j}j∈J−I,i∈[m] is a K − g−fusion woven for R(K).

Proof. The upper bound is obvious. Suppose that σi i∈[m] ⊂ J− I and f ∈ R(K), so we get∑
i∈[m]

∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉 =

∑
i∈[m]

∑
j∈σi∪I

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉 −

∑
i∈[m]

∑
j∈I

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉

≥ A〈K∗f , K∗f 〉 −
∑
i∈[m]

∑
j∈I

v2
i j‖Λi jPWi j

‖2〈f , f 〉

≥ (A− C‖K+‖2)〈K∗f , K∗f 〉.

�

Theorem 2.8. Let {Wi j ,Λi j , vi j}j∈J,i∈[m] be a K − g−fusion woven for H with bounds A,B. For
each i ∈ [m],j ∈ J and a index set Ii j , suppose that {f (k)

i j }k∈Ii j ∈ Λi j(Wi j) is a Parseval frame for
Hi j such that for every finite subset Ki j ⊂ Ii j , the set {f kij }k∈Ii j−Ki j is a frame with the lower bound
Ci j . Let W̃i j = span{Λ∗i j f

(k)
i j }k∈Ii j−Ki j for any i ∈ [m] and j ∈ J, then {W̃i j ,Λi j , vi j}j∈J,i∈[m] is a

K − g−fusion woven for H with the bounds (mini∈[m],j∈J Ci j)A and B.

Proof. Obviously, B is the upper bound of {W̃i j ,Λi j , vi j}j∈J,i∈[m]. Assume that f ∈ H and {σi}i∈[m] ∈
J, so ∑

i∈[m]

∑
j∈σi

v2
i j 〈Λi jPW̃i j

f ,Λi jPW̃i j
f 〉 =

∑
i∈[m]

∑
j∈σi

v2
i j

∑
k∈Ii j

〈Λi jPW̃i j
f , f

(k)
i j 〉〈f

(k)
i j ,Λi jPW̃i j

f 〉

≥
∑
i∈[m]

∑
j∈σi

v2
i j

∑
k∈Ii j−Ki j

〈Λi jPW̃i j
f , f

(k)
i j 〉〈f

(k)
i j ,Λi jPW̃i j

f 〉

≥
∑
i∈[m]

∑
j∈σi

v2
i jCi j〈Λi jPWi j

f ,Λi jPWi j
f 〉

≥ ( min
i∈[m],j∈J

Ci j)
∑
i∈[m]

∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉

≥ ( min
i∈[m],j∈J

Ci j)A〈K∗f , K∗f 〉.

�

Theorem 2.9. Let {Wi j ,Λi j , vi j}j∈J is a K−g−fusion frame for H for each i ∈ [m]. Suppose that for
a partition collection of disjoint finite sets {δi}i∈[m] of J and for any ε > 0 there exists a partition
{σi}i∈[m] of the set J − ∪i∈[m]δi such that {Wi j ,Λi j , vi j}j∈(σi∪δi ),i∈[m] has a lower K − g−fusion
frame bound less than ε. Then {Wi j ,Λi j , vi j}j∈J,i∈[m] is not a woven.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.11 8

Proof. We can write J = ∪j∈NJj , where Jj are disjoint index sets. Assume that δ1j = ∅ for all
i ∈ [m] and ε = 1. Then, there exists a partition σi1i∈[m] of J such that {Wi j ,Λi j , vi j}j∈(σi1∪δi1),i∈[m]has a lower bound (also, optimal lower bound) less than 1. Thus, there is a f1 ∈ H such that∑

i∈[m]

∑
j∈(σi1∪δi1)

v2
i j 〈Λi jPWi j

f1,Λi jPWi j
f1〉 < 〈K∗f1, K∗f1〉.

Since ∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

f1,Λi jPWi j
f1〉 <∞,

so, there is a k1 ∈ N such that∑
i∈[m]

∑
j∈K1

v2
i j 〈Λi jPWi j

f1,Λi jPWi j
f1〉 < 〈K∗f1, K∗f1〉,

where, K1 = ∪i≥k1+1Jj . Continuing this way, for ε = 1
n and a partition {δni}i∈[m] of J1 ∪ ...∪ Jkn−1such that

δni = δ(n−1)i ∪ (σ(n−1)i ∩ (J1 ∪ ... ∪ Jkn−1))

for all i ∈ [m], there exists a partition {σni}i∈[m] of J − (J1 ∪ ... ∪ Jkn−1) such that
{Wi j ,Λi j , vi j}j∈(σni∪δni ),i∈[m] has a lower bound less than 1

n . Therefore, there is a fn ∈ H and
kn ∈ N such that kn > kn−1 and∑

i∈[m]

∑
j∈Kn

v2
i j 〈Λi jPWi j

fn,Λi jPWi j
fn〉 <

1

n
〈K∗fn, K∗f1〉,

where, Kn = ∪i≥kn+1Jj . Choose a partition {αi}i∈[m] of J, where αi = ∪j∈N{δj i} = δ(n+1)i ∪ (αi ∩
J − (J1 ∪ ... ∪ Jn)). Assume that {Wi j ,Λi j , vi j}j∈αi ,i∈[m] is a K − g−fusion frame for H with theoptimal lower bound A. Then, by the Archimedean Property, there exits a r ∈ N such that r > 2

A .Now, there exists a fr ∈ H such that∑
i∈[m]

∑
j∈αi

v2
i j 〈Λi jPWi j

fr ,Λi jPWi j
fr 〉 =

∑
i∈[m]

∑
j∈δ(r+1)i

v2
i j 〈Λi jPWi j

fr ,Λi jPWi j
fr 〉

+
∑
i∈[m]

∑
j∈αi∩J−(J1∪...∪Jr )

v2
i j 〈Λi jPWi j

fr ,Λi jPWi j
fr 〉

≤
∑
i∈[m]

∑
j∈(σr i∪δr i )

v2
i j 〈Λi jPWi j

fr ,Λi jPWi j
fr 〉

+
∑
i∈[m]

∑
j∈∪k≥r+1Jk

v2
i j 〈Λi jPWi j

fr ,Λi jPWi j
fr 〉

<
1

r
〈K∗fr , K∗fr 〉+

1

r
〈K∗fr , K∗fr 〉

< A〈K∗fr , K∗fr 〉

and this is a contradiction with the lower bound of A. �

https://doi.org/10.28924/ada/ma.3.11


Eur. J. Math. Anal. 10.28924/ada/ma.3.11 9

Corollary 2.10. Let {Wi j ,Λi j , vi j}j∈J,i∈[m] be a K − g−fusion woven for H. Then there exists a
collection of disjoint finite subsets {δi}i∈[m] of J and A > 0 such that for each partition {σi}i∈[m]

of the set J − ∪i∈[m]δi , some the family {Wi j ,Λi j , vi j}j∈(σi∪δi ),i∈[m] is a K − g−fusion frame for H
with the lower frame bound A.

Theorem 2.11. Let {Wi j ,Λi j , vi j}j∈J be a K−g−fusion frame for H with bounds Ai and Bi for each
i ∈ [m]. Suppose that there exists N > 0 such that for all i , k ∈ [m] with i 6= k , I ⊂ J and f ∈ H,∑
j∈I
〈(vi jΛi jPWi j

− vkjΛkjPWkj
)f , (vi jΛi jPWi j

− vkjΛkjPWkj
)f 〉 ≤ N min{

∑
j∈I

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉,∑

j∈I
v2
kj〈ΛkjPWkj

f ,ΛkjPWkj
f 〉}.

Then the family {Wi j ,Λi j , vi j}j∈J,i∈[m] is woven with universal bounds

A

(m − 1)(N + 1) + 1
and B,

where A =
∑
i∈[m] Ai and B =

∑
i∈[m]Bi .

Proof. Let {σi}i∈[m] be a partition of J and f ∈ H. Therefore,∑
i∈[m]

Ai〈K∗f , K∗f 〉
∑
i∈[m]

∑
j∈J

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉

=
∑
i∈[m]

∑
k∈[m]

∑
j∈σk

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉

≤
∑
i∈[m]

(∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉+

∑
k∈[m],k 6=i

∑
j∈σk

{v2
kj〈ΛkjPWkj

f ,ΛkjPWkj
f 〉

+ 〈(vi jΛi jPWi j
− vkjΛkjPWkj

)f , (vi jΛi jPWi j
− vkjΛkjPWkj

)f 〉}
)

≤
∑
i∈[m]

(∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉

+
∑

k∈[m],k 6=i

∑
j∈σk

(N + 1)v2
kj〈ΛkjPWkj

f ,ΛkjPWkj
f 〉
)

= {(m − 1)(N + 1) + 1}
∑
i∈[m]

(∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉
)
.

Thus, we get
A

(m − 1)(N + 1) + 1
〈K∗f , K∗f 〉 ≤

∑
i∈[m]

(∑
j∈σi

v2
i j 〈Λi jPWi j

f ,Λi jPWi j
f 〉
)
≤ B〈f , f 〉.

�

In next theorem we study a Paley-Wiener type perturbation for weaving K − g−fusion frames.

https://doi.org/10.28924/ada/ma.3.11


Eur. J. Math. Anal. 10.28924/ada/ma.3.11 10

Theorem 2.12. Let {Wj ,Λj , wj}j∈J and {Vj , θj , vj}j∈J be two K−g−fusion frames for H with frame
bounds A1, B1 and A2, B2, respectively. Suppose that there exist non-negative scalers µ and
0 ≤ λ < 1

2 such that ( 1
2 − λ)A1 > µ and for each f ∈ H,∑

j∈J
〈(wjΛjPWj

− vjθjPVj )f , (wjΛjPWj
− vjθjPVj )f 〉 ≤ λ

∑
j∈J
〈wjΛjPWj

f , wjΛjPWj
f 〉+ µ〈K∗f , K∗f 〉.

Then, {Wj ,Λj , wj}j∈J and {Vj , θj , vj}j∈J are K−g−fusion woven for H with universal frame bounds
( 1

2 − λ)A1 − µ and B1 + B2.

Proof. The upper frame bound is clear. For the lower frame bound, assume that σ ⊂ J and we get,by the arithmetic-quadratic mean, for any f ∈ H∑
j∈σ
w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉+

∑
j∈σc

v2
j 〈θjPVj f , θjPVj f 〉

=
∑
j∈σ

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉

+
∑
j∈σc
〈wjΛjPWj

f − (wjΛjPWj
− vjθjPVj )f , wjΛjPWj

f − (wjΛjPWj
− vjθjPVj )f 〉

≥
∑
j∈σ

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉+

1

2

∑
j∈σc

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉

−
∑
j∈σc
〈(wjΛjPWj

− vjθjPVj )f , (wjΛjPWj
− vjθjPVj )f 〉

=
1

2

∑
j∈J

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉+

1

2

∑
j∈σ

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉

−
∑
j∈σc
〈(wjΛjPWj

− vjθjPVj )f , (wjΛjPWj
− vjθjPVj )f 〉

≥
1

2

∑
j∈J

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉 −

∑
j∈σc
〈(wjΛjPWj

− vjθjPVj )f , (wjΛjPWj
− vjθjPVj )f 〉

≥
1

2

∑
j∈J

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉 − λ

∑
j∈J

w2
j 〈ΛjPWj

f ,ΛjPWj
f 〉 − µ〈K∗f , K∗f 〉

≥
(

(
1

2− λ)A1 − µ
)
〈K∗f , K∗f 〉.

This completes the proof. �

Declarations
Availablity of data and materialsNot applicable.

https://doi.org/10.28924/ada/ma.3.11


Eur. J. Math. Anal. 10.28924/ada/ma.3.11 11

Human and animal rightsWe would like to mention that this article does not contain any studies with animals and does notinvolve any studies over human being.
Competing interestOn behalf of all authors, the corresponding author states that there is no conflict of interest.
FundingsAuthors declare that there is no funding available for this article.
Authors’ contributionsThe authors equally conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the final manuscript.

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https://doi.org/10.28924/ada/ma.3.11
https://doi.org/10.1090/s0002-9939-06-08498-x
https://doi.org/10.1090/s0002-9947-1952-0047179-6
https://doi.org/10.1049/ji-3-2.1946.0074
https://doi.org/10.2307/2372552
https://doi.org/10.2307/2372552
https://doi.org/10.1142/S0219691308002458
https://doi.org/10.28924/2291-8639-19-2021-836
https://doi.org/10.1142/S1793557120500606
https://doi.org/10.1142/S1793557120500606
https://doi.org/10.2478/aupcsm-2018-0002
https://doi.org/10.52547/ijmsi.17.1.1
https://doi.org/10.52547/ijmsi.17.1.1
https://doi.org/10.1007/s11785-022-01229-4

	1. Introduction
	2. Woven K-g-fusion frames in Hilbert C-modules 
	Declarations
	References