View of Continuous weaving $g$-frames in Hilbert $C^{\ast}$-modules

Nadia Assila¹,M’hamed Ghiati²,Mohammed Mouniane³,Mohamed Rossafi⁴

1Laboratory of Partial Differential Equations, Spectral Algebra and Geometry, Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco

2,3Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco

4LaSMA Laboratory Department of Mathematics Faculty of Sciences, Dhar El Mahraz University Sidi Mohamed Ben Abdellah, P. O. Box 1796 Fez Atlas, Morocco

Email address: ¹nadia.assila@uit.ac.ma,²mhamed.ghiati@uit.ac.ma,

3mohammed.mouniane@uit.ac.ma, ⁴rossafimohamed@gmail.com

Abstract.

The present paper aims to study c-g-woven for Hilbert C^∗-modules, first, we give some definitions and fundamental properties which will be useful to introduce this notion. And also some of his properties are given. Finally, we discuss the perturbation forc-g-woven.

Keywords: Continuous woveng-frames,g-frames, Hilbert C^∗-modules.

1.Introduction and Preliminaries

Frames generalise orthonormal bases and were introduced by Duffin and Schaefer [7] in 1952 to analyse some deep problems in nonharmonic Fourier series by abstracting the fundamental notion of Gabor [10] for signal processing. In 2000, Frank-larson [8] introduced the concept of frames in HilbetC^∗−modules as a generalization of frames in Hilbert spaces. The basic idea was to consider modules overC^∗−algebras of linear spaces and to allow the inner product to take values in theC^∗−algebras [14]. A. Khosravi and B. Khosravi [12] introduced the fusion frames andg−frame theory in Hilbert C^∗-modules. Afterwards, A. Alijani and M. Dehghan consider frames with C^∗-valued bounds [2] in Hilbert C^∗-modules. N. Bounader and S.

1

Received: 18 January 2023; Accepted: 30 April 2023; Published: 5 May 2023

Kabbaj [5] and A. Alijani [1] introduced the∗-g-frames which are generalizations of g-frames in Hilbert C^∗-modules. In 2016, Z. Xiang and Y. Li [26] give a generalization of g−frames for operators in Hilbert C^∗-modules. Recently, Fakhr-dine Nhari et al. [15] introduced the concepts of g-fusion frame and K-g-fusion frame in Hilbert C^∗-modules. Bemrose et al. [4]

introduced a new concept of weaving frames in separable Hilbert spaces. This notion has potential applications in distributed signal processing and wireless sensor networks. Weaving Frames in Hilbert C^∗-Modules introduced by X. Zhao and P. Li [29]. For more on frame in HilbertC^∗-modules see [11, 18, 19, 20, 21, 22] and references therein.

We organize the rest of the paper as follows. We continue with this section to collect some definitions, and basic lemmas which will be used in next. In section 2 we introduce the notion of continuousg-woven in Hilbert C^∗-module and discuss some of their properties. In section 3 we discuss perturbation ofc-g-woven.

Throughout this paper, (Ω, µ) is a measure space with positive measureµ,Hand{H_ω}_ω∈Ω are in HilbertC^∗-module and a family of HilbertC^∗-module, respectively, andU ∈End^∗_A(H,K) is the set of all adjointable operators from H into K. If H = K, then End^∗_A(H,H) will be denoted by H. For each m > 1 where m ∈ N, we define [m] := {1,2, . . . , m} and [m]^c={m+ 1, m+ 2, . . .}.

Definition 0.1. [16] LetA be a unitalC^∗-algebra and Hbe a leftA-module, such that the linear structures of A and H are compatible. His a pre-hilbert A-Module if His equipped with an A-valued in product ⟨., .⟩_A:H × H → A such that is sesquilinear, positive definite and respects the module action. In the other words,

(i) ⟨x, x⟩_A≥0 for allx∈ H and ⟨x, x⟩_A = 0 if and only if x= 0.

(ii) ⟨ax+y, z⟩_A =a⟨x, z⟩_A+⟨y, z⟩_A for all a∈ Aand x, y, z ∈ H.

(iii) ⟨x, y⟩A =⟨y, x⟩^∗_A for all x, y∈ H.

For x ∈ H we define ∥x∥= ∥⟨x, x⟩_A∥¹². If H is complete with ∥.∥, it is called a Hilbert A- module or a Hilbert C^∗-module overA.

For every ain C^∗-algebra A, we have |a|= (a^∗a)¹² and the A-valued norm on H is defined by |x|= (x^∗x)¹² forx∈ H.

LetHand K be tow Hilbert Amodules, AmapT :H → Kis said to be adjointable if there exists a mapT^∗:K → Hsuch that ⟨T x, y⟩_A =⟨x, T^∗y⟩_A for all y∈ K and x∈ H.

Definition 0.2. [27] Let T ∈ End^∗_A(H,K), then a bounded adjointable operator T^† ∈ End^∗_A(H,K) is called the Moore–Penrose inverse of T if T T^†T =T, T^†T T^†=T^†, T T^†∗

= T T^† and T^†T∗

=T^†T.

The notation T^† is reserved to denote the Moore-Penrose inverse of T. These properties imply thatT^†is unique andT^†T andT T^†are orthogonal projections. Moreover, Ran T^†

= Ran T^†T

,Ran(T) = Ran T T^†

,Ker(T) = Ker T^†T

and Ker T^†

= Ker T T^† which lead us toK = Ker T^†T

⊕Ran T^†T

= Ker(T)⊕Ran T^†

andH= Ker T^†

⊕Ran(T).

In [27] Xu and Sheng showed that a bounded adjointable operator between two Hilbert C^∗ modules admits a bounded Moore-Penrose inverse if and only if the operator has closed range, we refer the readers to [9, 23, 24] for more detailed information.

Lemma 0.3. [16] Let H and K two Hilbert A-module and T ∈ End^∗_A(H,K). Then, the following assertions are equivalent:

(i) The operator T is bounded andA-linear,

(ii) There exist k >0 such that ⟨T x, T x⟩_A ≤k⟨x, x⟩_A for allx∈ H.

Lemma 0.4. [3]. Let H and K be 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 isτ >0such that∥T^∗x∥ ≥τ∥x∥, for all x∈ K.

(iii) T^∗ is bounded below with respect to the inner product, i.e., there is ζ > 0 such that

⟨T^∗x, T^∗x⟩_A≥ζ⟨x, x⟩_A, for allx∈ K.

Lemma 0.5. [28]Let(Ω, µ)be a measure space,XandY are tow Banach spaces,λ:X →Y be a bounded linear operator and f :Ω→Y measurable function, then

λ(

Z

Ω

f dµ) = Z

Ω

(λf)dµ.

The following definitions were introduced by M. Rossafi et al. in the paper entitled ”Integral frame in Hilbert C^∗-module” (see [17]). Let (Ω, µ) be a measure space, H and V be two Hilbert C^∗-modules over a unital C^∗-algebra and {H_w}_w∈Ω is a family of submodules of End^∗_A(H,H_w) is the collection of all adjointableA-linear maps fromH intoH_w.

We define, following:

l²(Ω,{H_w}_ω∈Ω) =

F ={F_w}_w∈Ω :F_w ∈ H_w, Z

Ω

|F_w|²dµ(w)

<∞

.

For any F = {F_w}_w∈Ω and G = {G_w}_w∈Ω, the A-valued inner product is defined by

⟨F, G⟩ = R

Ω⟨F_w, G_w⟩_Adµ(w) and the norm is defined by ∥F∥ = ∥⟨F, F⟩∥¹². In this case thel²(Ω,{H_w}ω∈Ω) is an HilbertC^∗-module (see [14]).

Definition 0.6([13]). A family Λ :={Λ_ω∈End^∗_A(H,H_ω)}ω∈Ωis called a continuousg-frame (or briefly c−g−frame) in Hilbert A moduleH with respect to{H_ω}_ω∈Ω if

(i) the mapping

Ω7−→ A

ω7−→ ⟨Λ_ωf,Λωf⟩_A is measurable for anyf ∈ H.

(ii) there exist constants 0< A≤B <+∞ such that for eachf ∈ H,

(1) A⟨f, f⟩_A ≤

Z

Ω

⟨Λ_ωf,Λωf⟩_Adµ(ω)≤B⟨f, f⟩_A.

IfA,B can be chosen such thatA=B,then{Λ_ω}_ω∈Ω is called a tightc−g−frame and if A=B = 1,it is called parseval c-g-frame. A family {Λ_ω}_ω∈Ω is calledc−g−Bessel family if the right hand inequality (1) holds and the numberB is called the Bessel constant.

The continuousg−frame operator S_Λ on His defined by : SΛ(f) =

Z

Ω

Λ^∗_wΛwf dµ(w), f ∈ H.

Theorem 0.7. Let{Λ_ω}_ω∈Ω be a continuousg−Bessel family inHwith respect to{H_ω}_ω∈Ω with the bound B. Then the mappingTΛ on l²(Ω,{H_w}_ω∈Ω) to His defined by

⟨T_ΛF, g⟩_A = Z

Ω

⟨Λ^∗_ωF(ω), g⟩_Adµ(ω), F ∈l²(Ω,{H_w}_ω∈Ω), g∈ H, is linear and bounded with∥T_Λ∥≤√

B. Furthermore, for each g∈ H and ω∈Ω, T_Λ^∗(g)(ω) = Λ_ω(g).

In the continuous g− frame operatorSΛ=TΛT_Λ^∗ is defined by SΛ:H −→ H,

⟨S_Λf, g⟩_A= Z

Ω

⟨f,Λ_ωΛ^∗_ωg⟩_Adµ.

Therefore,

AI ≤SΛ≤BI

and we obtain, if {Λ_ω}_ω∈Ω is a c−g− frame, thenSΛ is positive, self-adjoint and invertible operator.

Proof. For any F ∈l²(Ω,{H_w}_ω∈Ω)

∥T_ΛF∥= sup

∥g∥=1

∥⟨T_ΛF, g⟩_A∥= sup

∥g∥=1

Z

Ω

⟨Λ^∗_ωF(w), g⟩_Adµ

= sup

∥g∥=1

Z

Ω

⟨F(w),Λ_ωg⟩_Adµ

≤ sup

∥g∥=1

Z

Ω

⟨F, F⟩_Adµ

1

2

Z

Ω

⟨Λ_ωg,Λ_ωg⟩_A dµ

1 2

≤√

B∥F∥.

Next, we show thatS_Λ is a self-adjoint operator

⟨S_Λf, g⟩_A =⟨T_ΛT_Λ^∗f, g⟩_A=⟨f, T_ΛT_Λ^∗g⟩_A=⟨f, S_Λg⟩_A Or, by using the notation from opertor theory, we get

AI ≤S_Λ≤BI,

thusSΛ is a positive operator furthermore

0≤I−B⁻¹S_Λ≤ B−A B I, and consequently

∥I−B⁻¹SΛ∥= sup

∥f∥=1

I−B⁻¹SΛf, f

A

≤ B−A B ≤1,

which shows thatSΛ is inverstible. □

The following Definition is a generalization of the Definition 3.1 in [25], for Hilbert C^∗- modules.

Definition 0.8. A family of c-frames {F_ωi}_{ω∈Ω,i∈[m]} in Hilbert C^∗-module H with respect toµ is said to be c-woven if there exist universal same positive constants 0 < A≤B <∞ such that for each partition{σ_i}_i∈[m]of Ω, the family{F_ωi}_{ω∈σi,i∈[m]}is ac−frame in Hilbert C^∗-moduleHwith bounds Aand B. Each family {F_ωi}_{ω∈σi,i∈[m]} is called a weaving.

2. Continuous weaving g-frames

In this section, we introduce the notation of continuous g-woven in Hilbert C^∗-module and discuss some of their properties.

Definition 0.9. Two c-g-frames Λ := {Λ_ωi ∈ End^∗_A(H,H_ω)}_{ω∈Ω,i∈[m]} and Γ := {Γ_ωi ∈ End^∗_A(H,H_ω)}_{ω∈Ω,i∈[m]} are said to be continuous g-woven (or c−g−woven) if there exist universal constants 0< A ≤ B <∞ such that for each partition {σ_i}_i∈[m] of Ω, the family {Λ_ωi}_{ω∈σi,i∈[m]}∪ {Γ_ωi}_{ω∈σi,i∈[m]} is a c−g− frame in Hilbert C^∗-module H with bounds A and B, respectively, that is

(2) A⟨f, f⟩A≤ Z

σ

⟨Λω_if,Λω_if⟩Adµ(ω) + Z

σ^c

⟨Γω_if,Γω_if⟩Adµ(ω)≤B⟨f, f⟩A.

In the above definition,A and B are called universal c-g-frames bounds.

It is easy to show that everyc-g-woven has an universal upper c-g-frame bound.

Indeed, let {Λ_ω}_ω∈Ω is be a c-g-Bessel family in Hilbert C^∗-module H with bounds B_i for each i∈[m]. Then, for any partition {σ_i}_i∈[m] of Ω and f ∈ H, we have

X

i∈[m]

Z

σi

⟨Λ_ωif,Λωif⟩Adµ≤ X

i∈[m]

Z

Ω

⟨Λ_ωif,Λωif⟩Adµ≤(X

i∈[m]

Bi)⟨f, f⟩A.

In the next results, we construct a c-g-woven by using a bounded linear operator.

Theorem 0.10. Let {Λ_ωi}_{ω∈Ω,i∈[m]} be a c-g- woven in Hilbert C^∗-module H with universal bounds A, B. If T ∈ End^∗_A(H) has closed range, then {Λ_ωiT^∗}_{ω∈Ω,i∈[m]} is a c−g− woven for R(T) with frame bounds A∥T^†∥⁻² andB∥T∥².

Proof. First, since T^∗f ∈ H and ω 7−→ ⟨Λ_ωif,Λ_ωif⟩_A, is measurable for any f ∈ H and i ∈ [m],then ω 7−→ ⟨Λ_ωiT^∗f,Λ_ωiT^∗f⟩_A, is measurable for any f ∈ H and i ∈ [m]. On the other hand, for eachf ∈ R(T), we have

A∥⟨f, f⟩_A∥=A

⟨T T^†f, T T^†f⟩_A

=A∥T T^†f∥²

=A∥(T^†)^∗T^∗f∥²

≤A∥(T^†)∥²∥T^∗f∥²

≤A∥(T^†)∥²∥⟨T^∗f, T^∗f⟩_A∥

≤ ∥(T^†)∥²

X

i∈[m]

Z

Ω

⟨Λ_ωiT^∗f,ΛωiT^∗f⟩_Adµ .

So,

A∥(T^†)∥⁻²∥⟨f, f⟩_A∥ ≤

X

i∈[m]

Z

Ω

⟨Λ_ωiT^∗f,ΛωiT^∗f⟩_Adµ

≤ X

i∈[m]

B_i∥⟨T^∗f, T^∗f⟩_A∥

≤B∥T∥²∥⟨f, f⟩_A∥.

□ The following result presents a relationship between the norms of thec-g-frame operator of the originalc-g-frame and the weaving.

Theorem 0.11. Let {Λ_ωi}_{ω∈Ω,i∈[m]} be a c-g-woven in H with universal boundsA and B. If S_Λ⁽ⁱ⁾ is the c-g-frame operator of {Λ_ωi}_ω∈Ω for each i ∈ [m], S_Λ,σi represents the c-g-frame operator of {Λ_ωi}_{ω∈Ω,i∈[m]} for each partition {σ_i}_i∈[m] of and S⁽ⁱ⁾_Λ,σ

i denotes the c-g-frame operatorS_Λ⁽ⁱ⁾ with integral restricted to {σ_i}, then for each f ∈ H,

X

i∈[m]

∥⟨S_Λ,σ⁽ⁱ⁾

if, S_Λ,σ⁽ⁱ⁾

if⟩_A∥≤B∥S_Λ,σi∥∥⟨f, f⟩_A∥.

Proof. Suppose that f ∈ H. We can write X

i∈[m]

∥⟨S_Λ,σ⁽ⁱ⁾

if, S_Λ,σ⁽ⁱ⁾

if⟩_A∥= X

i∈[m]

∥S_Λ,σ⁽ⁱ⁾

if∥²

= X

i∈[m]

( sup

∥g∥=1

∥⟨S_Λ,σ⁽ⁱ⁾

if, g⟩_A∥)²

= X

i∈[m]

( sup

∥g∥=1

∥⟨T_Λ,σ⁽ⁱ⁾

i(T_Λ,σ⁽ⁱ⁾

i)^∗f, g⟩_A∥)²

≤ X

i∈[m]

B∥⟨(T_Λ,σ⁽ⁱ⁾

i)^∗f,(T_Λ,σ⁽ⁱ⁾

i)^∗f⟩_A∥

=B X

i∈[m]

∥ Z

σi

⟨Λ_ωif,Λωif⟩_Adµ(ω)∥

≤B∥⟨S_Λ,σif, f⟩_A∥

≤B∥S_Λ,σi∥∥⟨f, f⟩A∥.

□

Theorem 0.12. Let Ω_i ⊆ Ω be measurable subsets for all i∈[m], and let F_i and G_i be c-frame mappings on Ω_i for H_ω with the pair frame bounds (A_Fi, B_Fi) and (A_Gi, B_Gi) re- spectively, for each ω ∈ Ω. Assume that Λ_ωi,Θ_ωi ∈End^∗_A(H,Ω_i), for any i∈[m] such that {Λ_ωi}_{ω∈Ω,i∈[m]} and {Θ_{ω∈Ω,i∈[m]} are strongly measurable. Then the following assertions are equivalent.

(I) {Λ^∗_ω

iFi}_{ω∈Ω,i∈[m]} and {Θ^∗_ω

iGi}_{ω∈Ω,i∈[m]} arec− woven in Hilbert C^∗-module H.

(II) {Λ_ωi}_{ω∈Ω,i∈[m]} and{Θ_ωi}_{ω∈Ω,i∈[m]} are c−g− woven in Hilbert C^∗-module H.

Proof. (I)⇒(II). Suppose thatσ⊆Ωis measurable subsets andf ∈ H. Let{Λ^∗_ω

iF_i}_{ω∈Ω,i∈[m]}

and{Θ^∗_ω

iGi}_{ω∈Ω,i∈[m]}arec-woven inH, with universal frame boundsC, DandA= inf{A_Fi, AGi}.

Then, for eachi∈[m],we have

A Z

σ

⟨Λ_ωif,Λ_ωif⟩_Adµ(ω) +A Z

σ^c

⟨Θ_ωif,Θ_ωif⟩_Adµ(ω)

≤ Z

σ

A_Fωi⟨Λ_ωif,Λωif⟩_Adµ(ω) + Z

σ^c

A_Gωi⟨Θ_ωif,Θωif⟩_Adµ(ω)

≤ Z

σ

Z

Ωi

|⟨Λ_ωif, Fi(x)⟩_A|²dµ(x)dµ(ω) + Z

σ^c

Z

Ωi

|⟨Θ_ωif, Gi(x)⟩_A|²dµ(x)dµ(ω)

= Z

σ

Z

Ωi

|⟨f,Λ^∗_ωiFi(x)⟩_A|²dµ(x)dµ(ω) + Z

σ^c

Z

Ωi

|⟨f,Θ^∗_ωiGi(x)⟩_A|²dµ(x)dµ(ω)

≤D⟨f, f⟩_A.

With the same way, we conclude that

B Z

σ

|⟨Λω_if,Λω_if⟩A|²dµ(ω) +B Z

σ^c

|⟨Θω_if,Θω_if,⟩A|²dµ(ω)

≥ Z

σ

Z

Ωi

|⟨f,Λ^∗_ωiFi(x)⟩_A|²dµ(x)dµ(ω) + Z

σ^c

Z

Ωi

|⟨f,Θ^∗_ωiGi(x)⟩_A|²dµ(x)dµ(ω)

≥C⟨f, f⟩A,

whereB= sup

BF_ωi, BG_ωi .Thus, we obtain that{Λω_i}_{ω∈Ω,i∈[m]}and{Θω_i}_{ω∈Ω,i∈[m]}arec-g-woven forH with universal frame bounds C

B and D A.

(II) ⇒ (I). Suppose that {Λω_i}_{ω∈Ω,i∈[m]} and {Θω_i}_{ω∈Ω,i∈[m]} are c-g-woven for H with universal

frame boundsCand D. Now, we can write for eachf ∈ H,

Z

σ

Z

Ω_i

|⟨f,Λ^∗_ωiFi(x)⟩A|²dµ(x)dµ(ω) + Z

σ^c

Z

Ω_i

|⟨f,Θ^∗_ωiGi(x)⟩A|²dµ(x)dµ(ω)

= Z

σ

Z

Ω_i

|⟨Λ_ωif, F_i(x)⟩_A|²dµ(x)dµ(ω) + Z

σ^c

Z

Ω_i

|⟨Θ_ωif, G_i(x)⟩_A|²dµ(x)dµ(ω)

≥ Z

σ

AF_ωi⟨Λω_if,Λω_if⟩_Adµ(ω) + Z

σ^c

AG_ωi⟨Θω_if,Θω_if⟩_Adµ(ω)

≥A Z

σ

⟨Λωif,Λ_ωif⟩Adµ(ω) + Z

σ^c

⟨Θωif,Θ_ωif⟩Adµ(ω)

≥AC⟨f, f⟩A.

Also, we can get

Z

σ

Z

Ωi

⟨f,Λ^∗_ω

iFi(x)⟩_Adµ(x)dµ(ω) + Z

σ^c

Z

Ωi

⟨f,Θ^∗_ω

iGi(x)⟩_Adµ(x)dµ(ω)≤BD⟨f, f⟩_A

So,

Λ^∗_ωiFi ω∈Ω,i∈[m] and

Θ^∗_ωiGi ω∈Ω,i∈[m] arec-woven forH, with universal boundsAC andBD.

□

Theorem 0.13. Let {Λ_ωi}_ω∈Ω be a c-g-frame in Hilbert C^∗-moduleH with frame bounds Ai

andBi for eachi∈[m]. Suppose that there exists M >0 such that for allf ∈ H, i̸=k∈[m]

and all measurable subset ∆⊂Ω

Z

∆

⟨(Λ_ωi−Λ_ωk)f,(Λ_ωi−Λ_ωk)f⟩_Adµ(ω)

≤Mmin

Z

∆

⟨Λ_ωif,Λ_ωif⟩_Adµ(ω) ,

Z

∆

⟨Λ_ωkf,Λ_ωkf⟩_Adµ(ω)

.

Then, the family{Λ_ωi}_{ω∈Ω,i∈[m]} is a c−g-woven with universal bounds

A

(m−1)(M+ 1) + 1 and B

where, A:=P

i∈[m]A_i andB =P

i∈[m]B_i.

Proof. The upper bound is evident. For the lower bound, suppose that{σ_i}_i∈[m]is a partition of Ω and f ∈ H.Therefore,

X

i∈[m]

Ai⟨f, f⟩_A

≤

X

i∈[m]

Z

Ω

⟨Λω_if,Λω_if⟩_Adµ(ω)

=

X

i∈[m]

X

k∈[m]

Z

σk

⟨Λω_if,Λω_if⟩_Adµ(ω)

≤

X

i∈[m]

Z

σi

⟨Λω_if,Λω_if⟩_Adµ(ω)

+ X

k∈[m]

Z

σk

{⟨Λω_if−Λω_kf,Λω_if −Λω_kf⟩dµ(ω) +⟨Λω_kf,Λω_kf⟩_Adµ(ω)}





≤

X

i∈[m]





 Z

σ_i

⟨Λωif,Λ_ωif⟩Adµ(ω) + X

k∈[m]

k̸=i

Z

σ_k

(M+ 1)⟨Λωkf,Λ_ωkf⟩Adµ(ω)







={(m−1)(M + 1) + 1}

X

i∈[m]

Z

σi

⟨Λω_if,Λω_if⟩_Adµ(ω) .

□ 3. Perturbation For C-G-Woven

Perturbation of frames has been discussed by Cazassa and Christensen in [6]. For weaving frames, Bemrose and et al. have studied in [4], also Vashisht and Deepshikha presented for continuous case in [25]. We aim to present it forc-g-woven.

Theorem 0.14. Suppose for each i∈[m], the family {Λ_ωi}_ω∈Ω is a c−g-frame in H with frame bounds A_i and B_i. Assume that there exist constants λ_i, η_i and γ_i(i∈ [m]) such that for some fixedn∈[m],

A:=An− X

i∈[m]\[n]

λi+ηi

pBn+γi

pBi

pBn+p Bi

>0 and

Z

Ω

Λ^∗_ωn−Λ^∗_ωi

F(w), g

Adµ

≤

ηi

Z

Ω

Λ^∗_ωnF(w), g

Adµ

+γi

Z

Ω

Λ^∗_ωiF(w), g

Adµ

+λi∥F∥ for everyF ∈l²(Ω,{H_w}_ω∈Ω)andg∈ H.Then for any partition{σ_j}_j∈[m]ofΩ,{Λ_ωi}_ω∈σ

j,j∈[m]

is ac-g-frame in HilbertC^∗-Module H with universal frame boundsA and P

i∈[m]Bi. Hence the family of c-g-frame {Λ_ωi}_{ω∈Ω,i∈[m]} is woven in HilbertC^∗-Module H.

Proof. It is clear that{Λ_ωif}_ω∈σ

j,j∈[m]is strongly measurable for eachf ∈ Hand any parti- tion {σ_j}_j∈[m] of Ω, also the family {Λ_ωi}_{ω∈Ω,i∈[m]} is a c-g-Bessel family with Bessel bound P

i∈[m]B_i. Now, we show that {Λ_ωi}_{ω∈Ω,i∈[m]} has the lower frame condition. Assume that TΛi is the synthesis operator of {Λ_ωi}_{ω∈Ω,i∈[m]}. Then for any F ∈ l²(Ω,{H_w}_ω∈Ω) for any, i∈[m]\{n}, we have

∥(T_Λn −T_Λi)F∥= sup

∥g∥=1

⟨(T_Λn−T_Λi)F(w), g⟩_A

= sup

∥g∥=1

⟨T_ΛnF(w)−T_ΛiF(w), g⟩_A

= sup

∥g∥=1

⟨T_ΛnF(w), g⟩_A− ⟨T_ΛiF(w), g⟩_A

= sup

∥g∥=1

F(w), T_Λ^∗_ng

A−

F(w), T_Λ^∗_ig

A

= sup

∥g∥=1

⟨F(w),Λωng⟩_A− ⟨F(w),Λωig⟩_A

= sup

∥g∥=1

Λ^∗_ωnF(w), g

A−

Λ^∗_ωiF(w), g

A

= sup

∥g∥=1

Z

Ω

Λ^∗_ωn−Λ^∗_ωi

F(w), g

Adµ

≤η_i sup

∥g∥=1

Z

Ω

Λ^∗_ωnF(w), g

Adµ

+γ_i sup

∥z∥=1

Z

Ω

Λ^∗_ωiF(w), g

Adµ

+λi∥F∥

=η_i∥T_ΛnF∥+γ_i∥T_ΛiF∥+λ_i∥F∥

≤ηi

pBn∥F∥+γi

pBi∥F∥+λi∥F∥

=

ηi

pBn+γi

pBi+λi

∥F∥.

Thus,

∥T_Λn−T_Λi∥ ≤η_ip

B_n+γ_ip

B_i+λ_i. (3)

For eachi∈[m] andσ ⊂Ω, we define

T_i^(σ): (⊕ω∈σHω, µ)_L2 →H D

T_i^(σ)G, hE

A = Z

σ

Λ^∗_ωiG(w), h

Adµ

for all G∈l²(Ω,{H_w}_ω∈Ω), we have

T_i^(σ)G

=∥T_Λi(G·χ_σ)∥

≤ ∥T_Λi∥ ∥G_·χσ∥

≤ ∥T_Λi∥ ∥F∥₂

≤p

Bi∥F∥₂

thus, T_i^(σ)

≤√

Bi, for eachi∈[m].Similarly with (3), we get for eachi∈[m]\{n},

T_n^(σ)−T_i^(σ) ≤ηi

pBn+γi

pBi+λi.

For everyf ∈H andi∈[m]\{n}, we compute

T_n^(σ) T_n^(σ)∗

−T_i^(σ)

T_i^(σ)∗ f

≤

T_n^(σ) T_n^(σ)∗

−T_n^(σ)

T_i^(σ)∗ f

+

T_i^(σ) T_i^(σ)∗

−T_n^(σ)

T_i^(σ)∗ f

≤ T_n^(σ)

T_n^(σ)∗

−

T_i^(σ)∗ f

+ T_i^(σ)

T_n^(σ)−T_i^(σ) f

≤ ηi

pBn+γi

pBi+λi

pBn+p Bi

∥f∥.

Now, suppose that{σ_i}_i∈[m]is a partition of Ω andTΛbe the synthesis operator associated with thec-g-Bessel family {Λ_ωi}_{ω∈Ω,i∈[m]}, we have

∥T_Λ^∗f∥²=∥⟨f, T_ΛT_Λ^∗f⟩_A∥

=

X

i∈[m]

Z

σi

f,Λ^∗_ωiΛ_ωif

Adµ(ω)

= Z

σ1

f,Λ^∗_ω1Λω1f

Adµ(ω) +. . .+ Z

σn

f,Λ^∗_ωnΛωnf

Adµ(ω) +. . . . . .+

Z

σm

f,Λ^∗_ωmΛωmf

Adµ(ω)

= Z

σ1

f,Λ^∗_ω1Λω1f

Adµ(ω) +. . .+ X

i∈[m]

Z

σi

f,Λ^∗_ωiΛωif

Adµ(ω)

− X

i∈[m]\{n}

Z

σi

f,Λ^∗_ωiΛ_ωif

Adµ(ω) + Z

σm

f,Λ^∗_ωmΛ_ωmf

Adµ(ω)

= Z

Ω

f,Λ^∗_ωnΛ_ωnf

Adµ(ω)−

X

i∈[m]\{n}

Z

σi

f,Λ^∗_ωnΛωnf

Adµ(ω) − Z

σi

f,Λ^∗_ωiΛωif

Adµ(ω)

≥ Z

Ω

f,Λ^∗_ωnΛωnf

Adµ(ω)

− X

i∈[m]\(n)

Z

σi

f,Λ^∗_ωnΛωnf

Adµ(ω)− Z

σi

f,Λ^∗_ωiΛωif

Adµ(ω)

≥A_n∥⟨f, f⟩∥_A− X

i∈[m]\{n}

D

T_n^(σ)

T_n^(σ) ∗

f, f E

A−D T_i^(σ)

T_i^(σ)

∗

f, f E

A

≥A_n∥⟨f, f⟩∥_A− X

i∈[m]\{n}

∥⟨f, f⟩∥_A

T_n^(σ)

T_n^(σ) ∗

−T_i^(σ)

T_i^(σ) ∗

≥A_n∥⟨f, f⟩∥_A− X

i∈[m]\{n}

∥⟨f, f⟩∥_A ηi

pBn+γi

pBi+λi

pBn+p Bi

=A⟨f, f⟩_A.

□ Corollary 0.15. For each i ∈ [m], let the family {Λ_ωi}_ω∈Ω be a c−g-frame in Hilbert C^∗-module H with frame bounds Ai and Bi. Assume that there exist constants λi, ηi, γi(i∈

[m−1]) andn∈[m] so that A:=A₁− X

i∈[m−1]\{n}

λ_i+η_ip

B_i+γ_ip

B_i+1 p

B_i+p B_i+1

>0 and

Z

Ω

D

Λ^∗_ωi−Λ^∗_ω(i+1)

F(w), g E

dµ

≤

ηi

Z

Ω

Λ^∗_ωiF(w), g

Adµ

+γi

Z

Ω

D

Λ^∗_ω(i+1)F(w), gE

Adµ

+λi∥⟨F, F⟩_A∥¹², for everyF ∈l²(Ω,{H_w}_ω∈Ω)andg∈ H.Then for any partition{σ_j}_j∈[m]ofΩ,{Λ_ωi}_ω∈σ

j,j∈[m]

is ac−g-frame in HilbertC^∗-ModuleHwith universal frame boundsAandP

i∈[m]Bi.Hence the family of c-g-frame {Λ_ωi}_{ω∈Ω,i∈[m]} is woven in HilbertC^∗-Module H.

Declarations

Availablity of data and materials Not applicable.

Competing interest

The authors declare that they have no competing interests.

Fundings

The authors declare that there is no funding available fot this paper.

Authors’ contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

References

[1] Alijani A, 2015.Generalized frames withC^∗-valued bounds and their operator duals, Filomat29:7, 1469–

1479. DOI 10.2298/FIL1507469A

[2] Alijani A, Dehghan M, 2011.∗-Frames in HilbertC^∗-modules, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys.73, no. 4, 89–106.

[3] Arambaˇsi´c L, 2006. On frames for countably generated HilbertC^∗-modules, Proc. Am. Math. Soc. 135 469–478.https://doi.org/10.1090/S0002-9939-06-08498-X.

[4] Bemrose T, Casazza P. G, Gr¨ochenig K, Lammers M. C, Lynch R. G, 2016. Weaving frames, Oper.

Matrices. 10, 1093–1116.https://doi.org/10.7153/oam-10-61.

[5] Bounader N, Kabbaj S, 2014.∗-g-frames in HilbertC^∗-modules, J. Math. Comput. Sci. 4, No. 2, 246-256.

[6] Cazassa P. G, Christensen O, 1997. Perturbation of Operators and Applications to Frame Theory, J.

Fourier Anal. Appl. 3.https://doi.org/10.1007/bf02648883.

[7] Duffin R. J, Schaeffer A. C, 1952. A Class of Nonharmonic Fourier Series, Trans. Amer. Math. Soc.

341-366.

[8] Frank M, Larson D. R, 2000.A-module frame concept for Hilbert C^∗-modules, functinal and harmonic analysis of wavelets, Contempt. Math. 247, 207-233.

[9] Frank M, Sharifi K, 2010. Adjointability of densely defined closed operators and theMagajna–Schweizer theorem, J. Operator Theory 63, 271–282.

[10] Gabor D, 1946.Theory of communications, J. Elect. Eng.93(1946), 429–457.

[11] Kabbaj S, Rossafi M, 2018. ∗-Operator frame for End^∗A(H), Wavelet Linear Algebra, 5, 1-13.

https://doi.org/10.22072/WALA.2018.79871.1153

[12] Khorsavi A, Khorsavi B, 2008.Fusion frames and g-frames in HilbertC^∗-modules, Int. J. Wavelet, Mul- tiresolution and Information Processing 6, 433-446.

[13] Kouchi M. R, Nazari A, 2011. Continuousg-frames in HilbertC^∗-modules, Hindawi Publishing Corpo- ration Abstract and Applied Analysis Volume, Article ID 361595, 20 pages.

[14] Lance E. C, 1995. HilbertC^∗-Modules: A Toolkit for Operator Algebraist, 144 pages, vol. 210 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK.

[15] Nhari F. D, Echarghaoui R, Rossafi M, 2021.K−g−fusion frames in HilbertC^∗−modules, Int. J. Anal.

Appl. 19 (6).

[16] Paschke W. L, 1973. Inner product modules over B∗-algebras, Trans. Am. Math. Soc. 182, 443–468.

https://doi.org/10.1090/s0002-9947-1973-0355613-0.

[17] Rossafi M, Chouchene F, Kabbaj S, 2020. Integral Frame in HilbertC^∗-Modules. https://arxiv.org/

abs/2005.09995v2

[18] Rossafi M, Nhari FD, Park C, Kabbaj S, 2022. Continuous g-Frames withC^∗-Valued Bounds and Their Properties. Complex Anal. Oper. Theory 16, 44. https://doi.org/10.1007/s11785-022-01229-4

[19] Rossafi M, Kabbaj S, 2020. ∗-K-operator frame for End^∗A(H), Asian-Eur. J. Math. 13, 2050060.

https://doi.org/10.1142/S1793557120500606.

[20] Rossafi M, Kabbaj S, 2019. Operator frame forEnd^∗A(H), J. Linear Topol. Algebra, 8, 85-95.

[21] Rossafi M, Kabbaj S, 2018.∗-K-g-frames in HilbertA-modules, J. Linear Topol. Algebra, 7, 63-71.

[22] Rossafi M, Kabbaj S, 2018.∗-g-frames in tensor products of HilbertC^∗-modules, Ann. Univ. Paedagog.

Crac. Stud. Math. 17, 17-25.

[23] Sharifi K, 2009. Descriptions of partial isometries on Hilbert C*-modules, Linear Algebra Appl. 431, 883–887.

[24] Sharifi K, 2010. Groetsch’s representation of Moore–Penrose inverses and ill-posed problems in Hilbert C*-modules, J. Math. Anal.Appl. 365, 646–652.

[25] Vashisht L. K, Deepshikha, 2017. On continuous weaving frames, Adv. Pure Appl. Math. 8, 15–31.

https://doi.org/10.1515/apam-2015-0077.

[26] Xiang Z, LI Y, 2016. G−frames for operators in Hilbert C^∗−modules, Turk J Math, 40, 453-469. Doi:

10.3906/mat-1501-22

[27] Xu Q, Sheng L, 2008. Positive semi-definite matrices of adjointable operators on HilbertC^∗-modules, Linear Algebra Appl. 428, 992–1000.https://doi.org/10.1016/j.laa.2007.08.035.

[28] Yosida K, 1980. Functional Analysis, vol. 123, Springer-Verlag Berlin Heidelberg, Springer, Berlin, Ger- many, 6th edition.

[29] Zhao X, Li P, 2021. Weaving Frames in HilbertC^∗-Modules, Journal of Mathematics, vol. 2021, Article ID 2228397, 13 pages.