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Duals of Continuous Frames in Hilbert C∗-Modules

Mohamed Rossafi1,∗, Khadija Mabrouk2, M’hamed Ghiati2, Mohammed Mouniane2

1Department of Mathematics Faculty of Sciences, Dhar El Mahraz University Sidi Mohamed Ben
Abdellah, Fez, Morocco

rossafimohamed@gmail.com
2Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco

khadija.mabrouk@uit.ac.ma, mhamed.ghiati@uit.ac.ma, mouniane.mohammed@uit.ac.ma
∗Correspondence: rossafimohamed@gmail.com

Abstract. The concept of frame is an exciting, dynamic, and fast-paced subject with applications innumerous fields of mathematics and engineering. The purpose of this paper is to introduce equiv-alent ∗-continuous frames and to present ordinary duals of constructed ∗-continuous frames by anadjointable and invertible operator. Also, we establish some properties.

1. Introduction
Frames in Hilbert spaces have been introduced by Duffin and Schaeffer [3] in 1952 to studysome deep problems in nonharmonic Fourier series. After the fundamental paper, by Daubechies,Grossman and Meyer [2], frame theory began to be widely used, particularly in the more specializedcontext of wavelet frame and Gabor frame [4]. Frames have been used in signal processing, imageprocessing, data compression and sampling theory. For more about frames, see [5, 7–11].In this paper, we introduce the notions of continuous frame on a Hilbert C∗-module over aunital C∗-algebra which is a generalization of discrete frame, the ∗-continuous frame, which are ageneralization of ∗-frame in Hilbert C∗-modules and we establish some new results.The paper is organized as follows. We continue this introductory section and briefly recall thedefinitions and basic properties of Hilbert C∗-modules. In Section 2, the generalized duals fora given ∗-continuous frame will be considered. Also, we study their properties and characterizeall operator dual ∗-continuous frames associated with the given ∗-continuous frame in Hilbert

C∗-modules. In Section 3, we extend this notion for sequences (continuous frames) in Hilbert
C∗-modules. Also, some properties of them will be studied. In Section 4, a ∗-continuous frame isconstructed by an orthogonal projection.

Received: 21 Jan 2024.2020 Mathematics Subject Classification. 42C15, 41A58.
Key words and phrases. Continuous frame; ∗-continuous frame; C∗-algebra; Hilbert C∗-module.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 2

Definition 1.1. [6] Suppose that A is a C∗-algebra. A linear space H which is also an algebraicleft A-module together with an A-inner product 〈·, ·〉 : H×H −→ A and possesses the followingproperties is called a pre-Hilbert C∗-module:(1) 〈f , f 〉 ≥ 0, for any f ∈ H;(2) 〈f , f 〉 = 0 if and only if f = 0;(3) 〈f , g〉 = 〈g, f 〉∗, for any f , g ∈ H;(4) 〈λf , h〉 = λ〈f , h〉, for any λ ∈ C and f , h ∈ H;(5) 〈af + bg, h〉 = a〈f , h〉+ b〈g, h〉, for any a, b ∈ A and f , g, h ∈ H.
For x ∈ H, we define ‖x‖ = ||〈x, x〉||

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 C∗-algebra A, we have |a| = (a∗a)

1
2 andthe A-valued norm on H is defined by |x | = 〈x, x〉

1
2 for x ∈ H. Let H and K be two Hilbert

A-modules. A map T : H → K is said to be adjointable if there exists a map T ∗ : K → H suchthat 〈Tx, y〉A = 〈x, T ∗y〉A for all x ∈ H and y ∈ K.
Lemma 1.2. [12] Let (Ω,µ) be a measure space, X and Y be two Banach spaces, λ : X → Y be
a bounded linear operator and f : Ω → Y be a measurable function. Then

λ(

∫
Ω

f dµ) =

∫
Ω

(λf )dµ.

Lemma 1.3. [1] Let H and K be two Hilbert A-modules and T ∈ End∗(H,K).(i) If T is injective and T has a 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.

The following definition was introduced in [11].
Definition 1.4. Let H be a Hilbert A-module and (Ω, µ) be a measure space. A map F : Ω→ His called a ∗-continuous frame with respect to (Ω, µ) if1. for all f ∈ H, w → 〈f , Fw 〉 is a measurable function on Ω,2. there exist two strictly nonzero elements A,B > 0 in A such that

A〈f , f 〉A∗ ≤
∫

Ω

〈f , Fw 〉〈Fw , f 〉dµ(w) ≤ B〈f , f 〉B∗,∀f ∈ H. (1.1)
The elements A and B are called ∗-continuous frame bounds. If A = B, we call this ∗-continuousframe a tight ∗-continuous frame, and if A = B = 1, it is called a Parseval ∗-continuous frame. Ifonly the right-hand inequality of (1.1) is satisfied, we call F : Ω→ H a ∗-continuous Bessel mapwith Bessel bound B.

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 3Let X be a Banach space, (Ω, µ) be a measure space and f : Ω→ X be a measurable function.Integral of the Banach-valued function f has been defined by Bochner and others. Most propertiesof this integral are similar to those of the integral of real-valued functions. Since every C∗-algebraand Hilbert C∗-module is a Banach space, we can use this integral and its properties.Let (Ω, µ) be a measure space. We define
L2(Ω,A) =

{
ϕ : Ω→ A :

∥∥∥∥∫
Ω

ϕ(ω)ϕ(ω)∗dµ(ω)

∥∥∥∥ <∞
}
.

For any ϕ,ψ ∈ L2(Ω,A), if the A-valued inner product is defined by
〈ϕ,ψ〉 =

∫
Ω

ϕ(ω)ψ(ω)∗dµ(w),

the norm is defined by ‖ϕ‖ = ‖〈ϕ,ϕ〉‖
1
2 , then L2(Ω,A) is a Hilbert C∗-module.The frame transform or pre-frame operator T : H −→ L2(Ω,A) is defined by T (f ) = {〈f , Fw 〉}w∈Ωand it is an injective and closed range adjointable A-module map and ‖T‖ ≤ ‖B‖. The adjointoperator T ∗ is surjective and it is given by T ∗ (ew ) = Fw for w ∈ Ω, where {ew}w∈Ω is thestandard basis for L2(Ω,A).

Definition 1.5. Let F be a continuous frame for H with respect to (Ω, µ). We define the frameoperator S : H → H by Sx = T ∗FTF x =
∫

Ω〈x, Fw 〉Fwdµ(w),∀x ∈ H, that is positive, invertibleand adjointable and the inequality ∥∥A−1
∥∥−2 ≤ ‖S‖ ≤ ‖B‖2

holds, and the reconstruction formula f =
∫

Ω

〈
f , S−1Fw

〉
Fwdµ(w) holds for all f ∈ H.

2. ∗-Continuous operator duals
Definition 2.1. Let {Fw}w∈Ω and {gw}w∈Ω be two ∗-continuous frames for H. If there exists aninvertible adjointable A-module map on H such that

x =

∫
Ω

〈Γx, gw 〉Fwdµ(ω), ∀x ∈ H, (2.1)
then {gw}w∈Ω is called a ∗-continuous operator dual of {Fw}w∈Ω.
Remark 2.2. Every ∗-continuous frame {Fw}w∈Ω with continuous frame operator S is a ∗-continuousoperator dual for itself. To see this, set Γ := S−1 and the reconstruction formula concludes it.
Remark 2.3. Every dual ∗-continuous frame {gw}w∈Ω of ∗-continuous frame {Fw}w∈Ω is a ∗-continuous operator dual when Γ = I, I is the identity operator on H.
Remark 2.4. Let G = {gw}w∈Ω be an operator dual of a ∗-continuous frame F = {Fw}w∈Ω in H.Then for some invertible adjointable map Γ ∈ B∗(H)

x =

∫
Ω

〈Γx, gw 〉Fwdµ(ω), ∀x ∈ H.

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 4The equality shows that I =
(
T ∗FTG

)
Γ, where I is the identity map on H, and TF and TG arepre-frame operators of F and G, respectively. Therefore, the operator Γ is unique and Γ−1 = T ∗FTG .

By Remark 2.4, we say that {gw}w∈Ω is an operator dual for {Fw}w∈Ω with the correspondingoperator Γ. Moreover, we mention that the operator duality relation of x-frames is symmetric. It isconsidered in the next remark.
Remark 2.5. If G = {gw}w∈Ω is an operator dual of a given ∗-continuous frame F = {Fw}w∈Ωwith the corresponding operator Γ, then {Fw}w∈Ω is an operator dual for {gw}w∈Ω with the cor-responding operator Γ∗. In order to see this, assume that TF and TG are pre-frame operators of Fand G, respectively. By the definition of operator duals, we have

I =

∫
Ω

〈Γx, gw 〉Fwdµ(ω) = (T ∗FTG) Γ.

Since Γ is invertible, Γ−1 = T ∗FTG and
I = Γ (T ∗FTG) =

(
T ∗GTF

)
Γ∗ =

∫
Ω

〈Γ∗f,Fw 〉 gwdµ(ω).

The following lemma is obtained by using the last remark and some properties of pre-frameoperators.
Lemma 2.6. Let F = {Fw}w∈Ω and G = {gw}w∈Ω be ∗-Bessel sequences for H with the pre-frame
operators TF and TG , respectively. Assume that Γ is an invertible and adjointable A-module map
on H. Then for x ∈ H, the following statements are equivalent:

(i) x =
∫

Ω 〈Γx, gw 〉Fwdµ(ω).
(i i) x =

∫
Ω 〈Γ

∗x, Fw 〉 gwdµ(ω).
In case that one of the above equalities is satisfied, {Fw}w∈Ω and {gw}w∈Ω are operator dual

∗-frames. Moreover, if B is an upper bound for {Fw}w∈Ω and S is frame operator of {Fw}w∈Ω,
then B

∥∥S−1
∥∥− 1

2 ‖TF‖−1 ‖Γ‖−1 is a lower bound for {gw}w∈Ω.

Proof. The equivalency of the two conditions is given from Remark 2.5.Now, let B be a ∗-Bessel bound for {Fw}w∈Ω and (i) holds. By the definition of ∗-Besselsequence {Fw}w∈Ω and T ∗FTGΓ = idH, we can write, for x ∈ H,
〈TFx, TFx〉 ≤ B〈x, x〉B∗ (2.2)

= B 〈T ∗FTGΓx, T ∗FTGΓx〉B∗ ≤ B ‖TF‖2 〈TGΓx, TGΓx〉B∗.

Using Lemma 1.3, we have∥∥∥(T ∗FTF)−1
∥∥∥−1
〈x, x〉 ≤ 〈TFx, TFx〉 , ∀f ∈ H. (2.3)

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 5It follows from Lemma 1.3, (2.2), and (2.3) that for x ∈ H,∥∥S−1
∥∥−1 ‖ΓΓ∗‖−1 〈f , f 〉 ≤

∥∥S−1
∥∥−1 〈

Γ−1f ,Γ−1f
〉
≤ B ‖TF‖2 〈TGf TGf 〉B∗,(

B−1
∥∥S−1

∥∥− 1
2 ‖TF‖−1 ‖Γ‖−1

)
〈f , f 〉

(
B−1

∥∥S−1
∥∥− 1

2 ‖TF‖−1 ‖Γ‖−1

)∗
≤ 〈TGf , TGf 〉 .

Therefore, B ∥∥S−1
∥∥∣∣− 1

2 ‖Γ‖−1 ‖TF‖−1 is a lower ∗-frame bound for {gj}j∈J and {gj}j∈J is a ∗-frame.Similarly, {fj}j∈J is also a ∗-frame. �

Proposition 2.7. Let
(
{gw}w∈Ω ,Γ

)
be an operator dual of a ∗-continuous frame {Fw}w∈Ω.1. For a strictly nonzero element α in the center of A, the pair

(
{αgw}w∈Ω , α

−1Γ
)

is an
operator dual for {Fw}w∈Ω.2. If Υ is an invertible and adjointable operator on H, then

(
{Υgw}w∈Ω , (Υ)−1Γ

)
is an

operator dual for {Fw}w∈Ω.3. The sequence {gw}w∈Ω is a dual of {Γ∗Fw}w∈Ω.4. Assume that
(
{hw}w∈Ω ,Λ

)
is another operator dual of {Fw}w∈Ω . Then(

{gw + hw}w∈J ,
(

Γ−1 + Λ−1
)−1
)

is an operator dual for {Fw}w∈Ω.

Proposition 2.8. Let
(
{gw}w∈Ω ,Γ

)
be an operator dual of {Fw}w∈Ω for H. If f is an element of

H such that 〈x, x〉 is a strictly nonzero element in the center of A, then
{〈
gw , (〈x, x〉)−1Γx

〉}
w∈Ω

is a dual of {〈Fw , x〉}w∈Ω.

Proof. Suppose that a ∈ A. Then∫
Ω

〈
a,
〈
gw , 〈x, x〉−1Γx

〉〉
〈Fw , x〉 dµ(ω) =

∫
Ω

a
〈
〈x, x〉−1Γx, gw

〉
〈Fw , x〉 dµ(ω)

= a〈x, x〉−1

〈∫
Ω

〈Γx, gw 〉Fwdµ(ω), x

〉
= a〈x, x〉−1〈x, x〉 = a.This completes the proof. �

Proposition 2.9. Let {Fw}w∈Ω be a ∗-continuous frame for H with frame operator S. If θ is an
adjointable and invertible operator on H, then

(
{θFw}w∈Ω ,

(
θ−1
)∗
S−1

)
is an operator dual for

{Fw}w∈Ω.

Proof. Let x ∈ H. Then∫
Ω

〈(
S−1θ−1

)
x, Fw

〉
θFwdµ(ω) = θ

(∫
Ω

〈(
S−1θ−1

)
xdµ(ω), Fw

〉
Fw

)
= θ

(
θ−1x

)
= x.

This completes the proof. �

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 6

Proposition 2.10. Let {Fw}w∈Ω be a ∗-continuous frame and θ be an adjointable and invertible
operator on H. Then the sets of operator duals of {Fw}w∈Ω and {θFw}w∈Ω are in one to one
correspondence.

Proof. First, suppose that ({gw}w∈Ω ,Γ
) is an operator dual for {Fw}w∈Ω. For x ∈ H, we obtain

x =

∫
Ω

〈Γ∗x, Fw 〉 gwdµ(ω) =

∫
Ω

〈
θ∗
(
θ−1
)∗

Γ∗x, Fw

〉
gwdµ(ω)

=

∫
Ω

〈(
θ−1
)∗

Γ∗x, θFw

〉
gwdµ(ω).

So ({gw}w∈Ω ,Γθ−1
) is an operator dual for {θFw}w∈Ω.Now, if ({gw}w∈Ω ,Γ
) is an operator dual of {θFw}w∈Ω, then ({gw}w∈Ω ,Γ∗θ

) is an operatordual of {Fw}w∈Ω, since
x =

∫
Ω

〈Γx, θFw 〉 gwdµ(ω) =

∫
Ω

〈θ∗Γx, Fw 〉 gwdµ(ω), ∀x ∈ H.

This completes the proof. �

Proposition 2.11. Let F = {Fw}w∈Ω be a ∗-continuous frame for H with pre-frame operator TF
and frame operator S. Then the set of all the operator duals of {Fw}w∈Ω is precisely the following

{gw}w∈Ω =

{
ΓFw + ϕew −

∫
Ω

〈
S−1Fw , Fi

〉
ϕewdµ(ω)

}
w∈Ω

,

where {ew}w∈Ω is the standard orthonormal basis for L2(Ω,A), ϕ ∈ B∗
(
H, L2(Ω,A)

)
, and Γ is

an invertible adjointable operator on H.

Proof. Assume that {gw}w∈Ω is a sequence as above. Then its pre-frame operator is TG = TFΓ +

ϕ− TFS−1T ∗Fϕ and so
(SΓ)−1 (T ∗FTG) = (SΓ)−1

(
T ∗FTFΓ + T ∗Fϕ− T ∗FTFS−1T ∗Fϕ

)
= (SΓ)−1

(
T ∗FTFS

−1SΓ + T ∗Fϕ− T ∗FTFS−1T ∗Fϕ
)

= (SΓ)−1(SΓ) = I.

By a similar relation with the given equality in Remark 2.5, we can conclude that {gw}w∈Ω isan operator dual for {Fw}w∈Ω with the corresponding operator (SΓ)−1. �

Theorem 2.12. Let
(
{gw}w∈Ω ,Γ

)
be an operator dual of ∗-continuous frame {Fw}w∈Ω for H. Then

there exist a Hilbert A-module K ⊇ H and a Riesz basis {uw}w∈Ω of K which has a unique dual
{vw}w∈Ω and satisfies (Pu)uw = Fw and (Pv0)vw = gw for all w ∈ Ω, where P is the projection
from K onto H.

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 7

Proof. Assume that TF , TG and SF , SG are pre-frame operators and frame operators of {Fw}w∈Ωand {gw}w∈Ω, respectively. Also, the orthogonal projections onto the range of TF , R (TF), and therange of TG , R (TG), are PF and PG , respectively. Now, for x ∈ H,〈
TGx, TGS

−1
G gw

〉
=
〈
T ∗GTGx, S

−1
G gw

〉
=
〈
S−1
G SGx, gw

〉
= 〈x, gw 〉 = 〈TGx, ew 〉 = 〈TGx, PGew 〉′and so

PGew = TGS
−1
G gw , ∀w ∈ Ω, (2.4)

where {ew}w∈Ω is the standard orthonormal basis of L2(Ω,A). By (2.4), for x ∈ H, we give
PGTF (Γ∗x) = PG

(∫
Ω

〈Γ∗x, Fw 〉 ew
)

=

∫
Ω

〈Γ∗x, Fw 〉PGew

=

∫
Ω

〈Γ∗F,Fw 〉TGS−1
G gw

= TGS
−1
G

(∫
Ω

〈Γ∗x, Fw 〉 gw
)

= TGS
−1
G x.

Set
K = H⊕ P⊥G l2(A), uw = Fw ⊕ P⊥G ew , ∀w ∈ Ω.

If TU is a pre-frame operator of the sequence {uw}w∈Ω, then TU(x ⊕ v) = TFx + v and
‖TU(x ⊕ v)‖ = ‖TF f + w‖ ≤ B(‖x‖+ ‖v‖) = B‖x ⊕ v‖, ∀x ⊕ v ∈ K

for some B > 0 and so {uw}w∈Ω is a Bessel sequence. We show that TU has a closed range.Suppose {ηn}n∈N ⊆ R (TU) such that ηn n→∞−→ η. Since Γ is invertible and adjointable, there exists
Γ∗Fn ⊕ vn ∈ H ⊕ P⊥G (L2(Ω,A);TU (Γ∗Fn ⊕ vn) = ηn. On the other hand,

TU (Γ∗Fn ⊕ vn) = TF (Γ∗Fn) + vn = ηn
n→∞−→ η

and Remark 2.4 gives
Fn = T ∗GTFΓ∗Fn = T ∗G (TFΓ∗fn + vn)

n→∞−→ T ∗Gη.

R (TU) is closed, since R (TF) is closed. Also, T ∗U has a closed range. This step will obtain theinjectivity of T ∗U . If T ∗U (∫Ω awew
)

= 0, then
0 =

∫
Ω

aw
(
Fw ⊕ P⊥G ew

)
=

∫
Ω

awFw ⊕ P⊥G
(∫

Ω

awew

)
.

It concludes that (i)
∫

Ω awFw = 0 and (i i) P⊥G
(∫

Ω awew
)

= 0. From (i i), we have∫
Ω

awew ∈ R (TG) =⇒ ∃h ∈ H;TGh =

∫
Ω

awew

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 8and on the other hand,
TGh =

∫
Ω

〈h, gw 〉 ew =⇒ aw = 〈h, gw 〉 , ∀w ∈ Ω.

From (i), we have
0 =

∫
Ω

awFw =

∫
Ω

〈h, gw 〉Fw =

∫
Ω

〈
ΓΓ−1h, gw

〉
Fw = Γ−1h.

Since Γ is injective, h = 0 and aw = 0 for w ∈ Ω. So ∫Ω awew = 0, and T ∗U is injective. Theoperator T ∗UTU is an invertible selfadjoint operator such that it has an upper bound and a lowerbound by Lemma 1.3, and also {uw}w∈Ω is a frame for K with frame operator SU = T ∗UTU . �

3. Equivalent ∗-continuous frames
Definition 3.1. Two sequences {Fw}w∈Ω and {gw}w∈Ω in H are said to be equivalent sequencesif there exists an adjointable and invertible operator Λ on H such that ΛFw = gw , for w ∈ Ω.
Theorem 3.2. Let {Fw}w∈Ω be a ∗-continuous frame for H and ξ be an adjointable and invertible
operator on H. Then every dual of the ∗-continuous frame {ξFw}w∈Ω is equivalent to a dual of
{Fw}w∈Ω, and the converse of the relation is valid.

Proof. First, suppose that {gw}w∈Ω is a dual of {Fw}w∈Ω. Then for x ∈ H, we obtain
x = ξ

(
ξ−1
)
x

= ξ

(∫
Ω

〈
ξ−1x, gw

〉
Fwdµ(ω)

)
=

∫
Ω

〈
x,
(
ξ−1
)∗
gw

〉
ξFwdµ(ω).

So {(ξ−1
)∗
gw
} is a dual for {ξFw}w∈Ω, and it is also equivalent to {gw}w∈Ω.Now, suppose that {hw}w∈Ω is a dual frame for {ξFw}w∈Ω. Set gw = ξ∗hw , for w ∈ Ω. Thenfor x ∈ H, ∫

Ω

〈x, gw 〉Fwdµ(ω) =

∫
Ω

〈x, ξ∗hw 〉 ξ−1ξFwdµ(ω)

= ξ−1

(∫
Ω

〈ξf , hw 〉 ξFwdµ(ω)

)
= ξ−1ξx

= x.Thus {gw}w∈Ω is a dual for {Fw}w∈Ω and hw =
(
ξ−1
)∗
gw . �

Theorem 3.3. If {Fw}w∈Ω and {gw}w∈Ω are ∗-continuous frames with the continuous frame opera-
tors SF and SG , respectively, then there exists a ∗-continuous frame that is equivalent to {gw}w∈Ω

and its frame operator is SF .

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 9

Proof. For the adjointable and invertible operator
ξ = S

1
2
FS
− 1

2
G ,

the sequence {ξgw}w∈Ω is a ∗-continuous frame with the frame operator Sξ = ξSGξ
∗. So

Sξ = ξSGξ
∗

=

(
S

1
2
FS
− 1

2
G

)
SG

(
S

1
2
FS
− 1

2
G

)∗
= SF .This completes the proof. �

Theorem 3.4. Let {Fw}w∈Ω and {gw}w∈Ω be ∗-continuous frames for H. Then the following
statements are valid.(1) {gw}w∈Ω is equivalent to a dual frame of {Fw}w∈Ω if and only if there exists an adjointable

and invertible operator ξ on H such that

ξf x =

∫
Ω

〈f , Fw 〉 gwdµ(ω), ∀x ∈ H.

(2) {gw}w∈Ω is equivalent to a dual frame of {Fw}w∈Ω if and only if there exists an adjointable
and invertible operator Γ such that

(
{gw}w∈Ω ,Γ

)
is an operator dual for {Fw}w∈Ω.

Proof. First, assume that {gw}w∈Ω is equivalent to a dual frame of {Fw}w∈Ω. Then there existsan adjointable and invertible operator Γ on H such that {Γgw}w∈Ω is a dual for {Fw}w∈Ω. Now,for x ∈ H,
=

∫
Ω

〈x, Fw 〉Γgwdµ(ω).

Set ξ = Γ−1. Then it concludes
ξx = Γ−1x

= Γ−1

(∫
Ω

〈x, Fw 〉Γgwdµ(ω)

)
=

∫
Ω

〈x, Fw 〉 gwdµ(ω).

In the second step, the adjointable and invertible operator ξ on H satisfies the following property
ξx =

∫
Ω

〈x, Fw 〉 gwdµ(ω), ∀x ∈ H.

Since ξ is invertible,
x =

∫
Ω

〈x, Fw 〉 ξ−1gwdµ(ω), ∀x ∈ H.

It shows that {ξ−1gw
}
w∈Ω

is a dual for {Fw}w∈Ω, and is equivalent to {gw}w∈Ω.For the proof of “if" part, assume that there exists a dual frame {hw}w∈Ω for {Fw}w∈Ω suchthat {hw}w∈Ω and {gw}w∈Ω are equivalent. Then there is an adjointable and invertible operator

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Eur. J. Math. Anal. 10.28924/ada/ma.4.4 10

Λ : H → H such that Λgw = hw for all w ∈ Ω. By Theorem 3.2, the sequence {gw}w∈Ω is a
∗-continuous frame.On the other hand, for x ∈ H,

x =

∫
Ω

〈x, hw 〉Fwdµ(ω)

=

∫
Ω

〈x,Λgw 〉Fwdµ(ω)

=

∫
Ω

〈Λ∗x, gw 〉Fwdµ(ω)

and so
x =

∫
Ω

〈Λ∗x, gw 〉Fwdµ(ω).This shows that {(gw ,Λ∗)} is an operator dual for {Fw}w∈Ω. The converse part is clear by thelast equalities. �

Theorem 3.5. Let {Fw}w∈Ω and {gw}w∈Ω be ∗-frames for H. Then {gw}w∈Ω is equivalent to an
operator dual frame of {Fw}w∈Ω if and only if there exists an adjointable and invertible operator
ξ on H such that

ξx =

∫
Ω

〈x, gw 〉Fwdµ(ω), ∀x ∈ H.

Proof. Suppose that {gw}w∈Ω is equivalent to {hw}w∈Ω, where ({hw}w∈Ω ,Γ
) is an operator dualfor {Fw}w∈Ω. Then there exists an adjointable and invertible operator θ on H such that θgw = hwfor all w ∈ Ω and for x ∈ H,

x =

∫
Ω

〈Γx, hw 〉Fw

=

∫
Ω

〈Γx, θgw 〉Fwdµ(ω)

=

∫
Ω

〈θ∗Γx, gw 〉Fwdµ(ω).

Set ξ = (θ∗Γ)−1. Then the result is obtained.For the converse, let ξ be an adjointable and invertible operator on H that
ξx =

∫
Ω

〈x, gw 〉Fwdµ(ω).

Then
x =

∫
Ω

〈
ξ−1x, gw

〉
Fwdµ(ω), ∀x ∈ H

and ({gw}w∈Ω , ξ
−1
) is an operator dual of {Fw}w∈Ω and {gw}w∈Ω is equivalent to itself. �

Moreover, some equivalence frames have the same Grammian matrices. These frames are intro-duced in the following proposition.
Proposition 3.6. Let {Fw}w∈Ω and {gw}w∈Ω be equivalent Parseval frames for H and let GF and
GG be Grammian matrices of {Fw}w∈Ω and {gw}w∈Ω, respectively. Then GF = GG .

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 11

Proof. Since two frames {Fw}w∈Ω and {gw}w∈Ω are equivalent, there exists an adjointable andinvertible operator ξ : H −→ H by ξFw = gw for w ∈ Ω. Since their frame operators are theidentity operator on H, by Theorem 3.2,
ξξ∗ = ξidξ∗ = id.

So ξ is a unitary operator and then for i , w ∈ Ω,
〈gi , gw 〉 = 〈ξFi , ξFw 〉 = 〈Fi , Fw 〉 ,

which shows that GF = [〈Fi , Fw 〉]w∈Ω = [〈gi , gw 〉]w∈Ω = GG . �

4. Constructed ∗-continuous frames and some properties
Theorem 4.1. Let {Fw}w∈Ω be a ∗-continuous frame for H and ξ be an adjointable and invertible
operator on H. Then the set

(
{gw}w∈Ω ,Γξ−1

)
is all of operator duals of {ξFw}w∈Ω, where(

{gw}w∈Ω ,Γ
)

is an operator dual for {Fw}w∈Ω.

Proof. Let ({gw}w∈Ω ,Γ
) be an operator dual of {Fw}w∈Ω. Then for x ∈ H,∫

Ω

〈
Γξ−1x, gw

〉
ξxwdµ(ω) = ξ

(∫
Ω

〈
Γξ−1f , gw

〉
Fwdµ(ω)

)
= ξ

(
ξ−1
)
x

= x.

This shows that ({gj}j∈J ,Γξ−1
) is an operator dual of {ξfj}j∈J .Now, if ({gj}j∈J ,Γ

) is an operator dual for {ξfj}j∈J , then it is enough to set Γ := Γξ in thelast equalities which follows that ({gj}j∈J ,Γ
) is an operator dual for {fj}j∈J . �

An orthogonal projection will obtain a ∗-frame, and relation will also be given for this projection.To see this, we must show that the inverse of the frame operator is unique in the reconstructionformula. So, firstly this fact will be considered.
Theorem 4.2. If {Fw}w∈Ω is a ∗-continuous frame for H, then there exists a unique adjointable
operator Λ on H such that

x =

∫
Ω

〈x,ΛFw 〉Fwdµ(ω), ∀x ∈ H.

Proof. By the reconstruction formula, there exists Λ = S−1. For the uniqueness of S−1 with thisproperty, we know that {S− 1
2Fw

}
w∈Ω

is a continuous Parseval frame for H. Set gw = S−
1
2Fw .Then Fw = S

1
2 gw . Now, suppose that Λ is an adjointable operator such that

x =

∫
Ω

〈x,ΛFw 〉Fwdµ(ω), ∀x ∈ H.

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 12Then we have
x =

∫
Ω

〈x,ΛFw 〉Fwdµ(ω)

=

∫
Ω

〈
x,ΛS

1
2 gw

〉
S

1
2 gwdµ(ω)

= S
1
2

(∫
Ω

〈
x,ΛS

1
2 gw

〉
gwdµ(ω)

)
= S

1
2

(∫
Ω

〈
S

1
2 Λ∗x, gw

〉
gw

)
= S

1
2

(
S

1
2 Λ∗x

)
= SΛ∗x, ∀x ∈ H.This concludes that SΛ∗ = id and then Λ∗ = S−1. More precisely, Λ is self-adjoint, positive andinvertible. �

Now, a ∗-continuous frame is constructed by an orthogonal projection.
Proposition 4.3. Let {Fw}w∈Ω be a ∗-continuous frame for H with the frame operator S and ∗-
continuous frame bounds A and B. Also, suppose that P is an orthogonal projection on H. Then
{PFw}w∈Ω is a ∗-continuous frame for RP with ∗-continuous frame bounds A and B. Moreover,
if
(
{gw}w∈Ω ,Γ

)
is an operator dual of {Fw}w∈Ω, then {PΓ∗gw}w∈Ω is a dual ∗-continuous frame

for {P fw}w∈Ω.

Proof. For x ∈ RP ,∫
Ω

〈x, PFw 〉 〈PFw , x〉 dµ(ω) =

∫
Ω

〈Px, Fw 〉 〈Fw , P x〉 dµ(ω)

=

∫
Ω

〈x, Fw 〉 〈Fw , x〉 dµ(ω)

and by the definition of ∗-continuous frame {Fw}w∈Ω, we have
A〈x, x〉A∗ ≤

∫
Ω

〈x, PFw 〉 〈PFw , x〉 dµ(ω) ≤ B〈x, x〉B∗.

Now, if ({gw}w∈Ω ,Γ
) is an operator dual of {Fw}w∈Ω, then for x ∈ RP ,

x = Px

= P

(∫
Ω

〈ΓPx, gw 〉Fwdµ(ω)

)
=

∫
Ω

〈x, PΓ∗gw 〉PFwdµ(ω).

If {gw}w∈Ω is also a dual of {Fw}w∈Ω, then {Pgw}w∈Ω is a dual of {PFw}w∈Ω. So the result isclear by Γ = idH. �

By the last theorem, a necessary and sufficient condition is found for commutating a projectionwith the inverse of the frame operator of a given ∗-frame.

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 13

Theorem 4.4. Let {Fw}w∈Ω be a ∗-continuous frame for H with the frame operator S. Suppose
that P is an orthogonal projection on H. Then PS−1Fw = S−1

P PFw , for all w ∈ Ω if and only if
PS−1 = S−1P , where SP is the continuous frame operator of the ∗-continuous frame {PFw}w∈Ω.

Proof. First, assume that PS−1Fw = S−1
P PFw , for all w ∈ Ω. Now, let x ∈ H. Then we have

S−1
P Px = S−1

P P

(∫
Ω

〈
x, S−1Fw

〉
Fwdµ(ω)

)
=

∫
Ω

〈
x, S−1Fw

〉
S−1
P PFwdµ(ω)

=

∫
Ω

〈
x, S−1Fw

〉
PS−1Fwdµ(ω)

= PS−1

(∫
Ω

〈
x, S−1Fw

〉
Fwdµ(ω)

)
= PS−1x.

Therefore, PS−1x = S−1
P Px , for all x ∈ H, and so

S−1
P P = PS−1P ⇒ PS−1P = PS−1.

Thus
PS−1P =

(
PS−1P

)∗
=
(
PS−1

)∗
= S−1P.For the proof of converse, suppose that PS−1 = S−1P . Let x ∈ RP . Then we have

x = Px

= P

(∫
Ω

〈
x, S−1Fw

〉
Fwdµ(ω)

)
=

∫
Ω

〈
x, S−1Fw

〉
PFwdµ(ω)

=

∫
Ω

〈
Px, S−1Fw

〉
PFwdµ(ω)

=

∫
Ω

〈
x, PS−1Fw

〉
PFwdµ(ω)

=

∫
Ω

〈
x, S−1PFw

〉
PFwdµ(ω).By Theorem 4.2 and the assumption, for w ∈ Ω,

S−1PFw = S−1
P Fw = S−1

P PFw = PS−1Fw

and the proof is complete. �

Theorem 4.5. Let {Fw}w∈Ω be a continuous Parseval frame of H with pre-frame operator θF and
let {gw}w∈Ω be a ∗-continuous frame with the pre-frame operator θG . Then

(
{gw}w∈Ω ,Γ

)
is an

operator dual for {Fw}w∈Ω if and only if PθF θGΓ = θF , where PθF is the orthogonal projection on
the range of θF .

https://doi.org/10.28924/ada/ma.4.4


Eur. J. Math. Anal. 10.28924/ada/ma.4.4 14

Proof. Suppose that ({gw}w∈Ω ,Γ
) is an operator dual for {Fw}w∈Ω. Since {Fw}w∈Ω is a contin-uous Parseval frame, the pre-frame operator θF is an isometry

〈θFx, θFx〉 =

∫
Ω

〈x, Fw 〉 〈Fw , x〉 dµ(ω) = 〈x, x〉, ∀x ∈ H.

Then for x ∈ H, 〈
PθF θGΓx, θFx

〉
=
〈
θGΓx, PθF θFx

〉
= 〈θGΓf , θFx〉

=

〈∫
Ω

〈Γx, gw 〉Fwdµ(ω), x

〉
= 〈x, x〉

= 〈θFx, θFx〉 .Thus PθF θGΓ = θF .Conversely, since θF is an isometry, by a similar method, we have
〈x, g〉 = 〈θFx, θFg〉

=
〈
PθF θGΓx, θFg

〉
=

〈∫
Ω

〈Γx, gw 〉Fwdµ(ω), g

〉
, ∀x ∈ H,

and so
x =

∫
Ω

〈Γx, gw 〉Fwdµ(ω), ∀x ∈ H.

Thus ({gw}w∈Ω ,Γ
) is a dual for {Fw}w∈Ω. �

Corollary 4.6. Let {Fw}w∈Ω and {gw}w∈Ω be two ∗-continuous frames for H with pre-frame
operators θF and θG , respectively. Then

(
{gw}w∈Ω ,Γ

)
is an operator dual for {Fw}w∈Ω if and

only if θ∗FPθF θGΓ = id .

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.
References

[1] A. Alijani, M.A. Dehghan, G-Frames and their duals in Hilbert C∗-modules, Bull. Iran. Math. Soc. 38 (2012),567–580.[2] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271–1283.[3] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.[4] D. Gabor, Theory of communications, J. Inst. Electr. Eng. 93 (1946), 429–457.[5] F. D. Nhari, R. Echarghaoui, M. Rossafi, K-g-fusion frames in Hilbert C∗-modules, Int. J. Anal. Appl. 19 (2021),836–857.[6] W. Paschke, Inner product modules over B∗-algebras, Trans. Am. Math. Soc. 182 (1973), 443–468.[7] M. Rossafi, S. Kabbaj, ∗-K-g-frames in Hilbert A-modules, J. Linear Topol. Algebra 7 (2018), 63–71.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.4 15

[8] M. Rossafi, S. Kabbaj, ∗-g-frames in tensor products of Hilbert C∗-modules, Ann. Univ. Paedagog. Crac. Stud. Math.17 (2018), 17–25.[9] M. Rossafi, S. Kabbaj, Operator frame for End∗A(H), J. Linear Topol. Algebra 8 (2019), 85–95.[10] M. Rossafi, S. Kabbaj, ∗-K-operator frame for End∗A(H), Asian-Eur. J. Math. 13 (2020), 2050060.[11] M. Rossafi, F. D. Nhari, C. Park, S. Kabbaj, Continuous g-frames with C∗-valued bounds and their properties,Complex Anal. Oper. Theory 16 (2022), 44.[12] K. Yosida, Functional analysis, vol. 123, Grundlehren der Mathematischen Wissenschaften, Springer, Berlin andNew York, 1980.

https://doi.org/10.28924/ada/ma.4.4

	1. Introduction
	2. -Continuous operator duals
	3. Equivalent *-continuous frames
	4. Constructed -continuous frames and some properties
	References

