Int. J. Anal. Appl. (2022), 20:24

K −g−Duals in Hilbert C∗−Modules

Mohamed Rossafi1, Fakhr-dine Nhari2,∗

1LaSMA Laboratory Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University

Sidi Mohamed Ben Abdellah, P. O. Box 1796 Fez Atlas, Morocco

2Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty of Sciences,

University of Ibn Tofail, P. O. Box 133 Kenitra, Morocco

∗Corresponding author: nharidoc@gmail.com

Abstract. Generalized frames with adjointable operators called K−g−frame is a generalization of a
g−frame. In this paper, we give some results of dual K−g−bessel sequence, finally we obtain a new
properties of approximate K−g−duals in Hilbert C∗−module.

1. introduction

Frames, introduced by Duffin and Schaefer [1] in 1952 to analyse some deep problems in nonhar-

monic Fourier series by abstracting the fundamental notion of Gabor [2] for signal processing. In 2000,

Frank-larson [4] introduced the concept of frames in Hilbet C∗−modules as 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 [5]. Many generalizations of the concept of
frame have been defined in Hilbert C∗-modules [3,6,8–10,12–15].

Throughout this paper, H is considered to be a countably generated Hilbert A−module. Let {Hj}j∈J
be the collection of submodules of H where I is a finite or countable index set. End∗A(H,Hj) is the

set of all adjointable operator from H to Hj. In particular End∗A(H) denote the set of all adjointable

operators on H.

Received: Feb. 28, 2022.

2010 Mathematics Subject Classification. Primary 41A58, Secondary 42C15.

Key words and phrases. g-Frame, K-g-Frame, C∗-algebra, Hilbert A-modules.

https://doi.org/10.28924/2291-8639-20-2022-24
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-24


2 Int. J. Anal. Appl. (2022), 20:24

Define the module

l2({Hj}j∈J) = {{xj}j∈J : xj ∈ Hj,‖
∑
j∈J
〈xj,xj〉‖ < ∞}

with A−valued inner product 〈x,y〉 =
∑
j∈J〈xj,yj〉, where x = {xj}j∈J and y = {yj}j∈J, clearly

l2({Hj}j∈J) is a Hilbert A−module.
In the following we briefly recall some definitions and basic properties.

For a C∗-algebra A if a ∈A is positive we write a ≥ 0 and A+ denotes the set of positive elements
of A.

Definition 1.1. [7]. Let A be a unital C∗-algebra and H be a left A-module, such that the linear
structures of A and U 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) 〈x,x〉≥ 0 for all x ∈ H and 〈x,x〉 = 0 if and only if x = 0.
(ii) 〈ax + y,z〉 = a〈x,z〉 + 〈y,z〉 for all a ∈A and x,y,z ∈ H.
(iii) 〈x,y〉 = 〈y,x〉∗ for all x,y ∈ 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 and the A-valued

norm on H is defined by |x| = 〈x,x〉
1
2 for x ∈ H.

Definition 1.2. Suppose that X and Y are Hilbert A−modules and T ∈ End∗A(X,Y ). The Moore-
Penrose inverse of T (if it exists) is an element T + of End∗A(Y,X) satisfying

TT +T = T, T +TT + = T +, (TT +)∗ = TT +, (T +T )∗ = T +T

Lemma 1.1. Let X and Y be two Hilbert A−modules and T ∈ End∗A(X,Y ) Then the Moore-Penrose
inverse T + exixts if and only if T has a closed range.

Definition 1.3. Suppose that K ∈ End∗A(H) and {Λj}j∈J is a K − g−frame for H. A g−bessel
sequence {Γj}j∈J for H is said to be a dual K −g−bessel sequence of {Λj}j∈J if

Kx =
∑
j∈J

Λ∗j Γjx, ∀x ∈ H.

Definition 1.4. [3] A sequence {Λj ∈ End∗A(H,Hj), j ∈ J} is called a g−frame with respect to
{Hj}j∈J if there exist constants C,D > 0 such that for every x ∈ H

C〈x,x〉≤
∑
j∈J
〈Λjx, Λjx〉≤ D〈x,x〉. (1.1)

As usual C and D are g−frame bounds of {Λj}j∈J. If only upper inequality of (1.1) holds, then {Λj}j∈J
is called g−bessel sequence for H.



Int. J. Anal. Appl. (2022), 20:24 3

Definition 1.5. Let K ∈ End∗A(H) and {Λj ∈ End
∗
A(H,Hj), j ∈ J}. A sequence {Λj}j∈J is called

K −g−frame for H with respect to {Hj}j∈J, if there exist constants 0 < A ≤ B < ∞ such that

A〈K∗x,K∗x〉≤
∑
j∈J
〈Λjx, Λjx〉≤ B〈x,x〉,∀x ∈ H.

A K −g−frame {Λj}j∈J is said to be tight if there exists a constant A > 0 such that

∑
j∈J
〈Λjx, Λjx〉 = A〈K∗x,K∗x〉,∀x ∈ H.

2. Main Results

Theorem 2.1. Suppose K ∈ End∗A(H) has closed range and {Λj ∈ End
∗
A(H,Hj)} is a K −g−frame

for H. for each j ∈ J, let Γj ∈ End∗A(H,Hj). Then the following statement are equivalent

(1) The sequence {Γj}j∈J is a dual K −g−bessel sequence of {Λj}j∈J.
(2) For each a ∈ Ker(K),

∑
j∈J Λ

∗
j Γja = 0. the sequence {Γj}j∈J is a g−bessel sequence

for H, which can naturally generate a g−bessel sequence {θj}j∈J for H in the forme θj =
ΓjK

+Prange(K) for each j ∈ J such that

h =
∑
j∈J

Λ∗j θjh =
∑
j∈J

θ∗j Λjh, ∀h ∈ range(K) (2.1)

where Prange(K) denotes the orthogonal projection onto range(K).

(1) =⇒ (2) Assume that {Γj}j∈J is a dual K − g−bessel sequence of {Λj}j∈J, then for each
a ∈ Ker(K),

∑
j∈J Λ

∗
j Γja = Ka = 0 and we have for any h ∈ range(K)

h = KK+h =
∑
j∈J

Λ∗j ΓjK
+h =

∑
j∈J

Λ∗j ΓjK
+Prange(K)h,

put θj = ΓjK+Prange(K), for each j ∈ J
so for any x ∈ H

∑
j∈J
〈θjx,θjx〉 =

∑
j∈J
〈ΓjK+Prange(K)x, ΓjK

+Prange(K)x〉

≤ DΓ〈K+x,K+x〉

≤ DΓ‖K+‖2〈x,x〉,

hence {θj}j∈J is g−bessel sequence for H.



4 Int. J. Anal. Appl. (2022), 20:24

Let g,h ∈ range(K)

〈g,h〉 = 〈
∑
j∈J

Λ∗j θjg,h〉

=
∑
j∈J
〈θjg, Λjh〉

=
∑
j∈J
〈ΓjK+Prange(K)g, Λjh〉

=
∑
j∈J
〈g, (ΓjK+Prange(K))

∗Λjh〉

= 〈g,
∑
j∈J

θ∗j Λjh〉,

therefore,
∑
j∈J θ

∗
j Λjh = h (2) =⇒ (1) We have range(K

+) = (Ker(K))⊥ then each x ∈ H can be
expressed as x = x1 + x2, where x1 ∈ range(K+) and x2 ∈ (range(K+))⊥ = Ker(K), so

Kx1 =
∑
j∈J

Λ∗j ΓjK
+Prange(K)Kx1

=
∑
j∈J

Λ∗j ΓjK
+Kx1

=
∑
j∈J

Λ∗j Γjx1.

And we have, Kx2 =
∑
j∈J Λ

∗
j Γjx2 = 0, we obtain

Kx = K(x1 + x2)

= Kx1

=
∑
j∈J

Λ∗j Γjx1

=
∑
j∈J

Λ∗j Γj(x1 + x2)

=
∑
j∈J

Λ∗j Γjx.

Hence, {Γj}j∈J is a dual K −g−bessel sequence of {Λj}j∈J.

Theorem 2.2. Suppose that K ∈ End∗A(H), Λj, Λ
′
j ∈ End

∗
A(H,Hj) for each j ∈ J and that {Γj ∈

End∗A(Hj,Wji )}i∈Ij is a g−frame for Hj with bounds Cj,Dj such that 0 < C = infj∈J Cj ≤ supj∈J Dj =
D < ∞.

Let {Γ
′
ji ∈ End

∗
A(Hj,Wji )}i∈Ij be a dual g−frame of {Γji}i∈Ij for each j ∈ J. Then the following

conditions are equivalent

(1) The pair {Λj}j∈J and {Λ
′
j}j∈J are a dual K −g−frame pair.

(2) The pair {Γji Λj}j∈J,i∈Ij and {Γ
′
ji Λ

′
j}j∈J,i∈Ij are a dual K −g−frame pair.



Int. J. Anal. Appl. (2022), 20:24 5

Proof. Suppose that {Λj}j∈J is a K −g−frame for H with bounds DΛ and CΛ.
For each x ∈ H, we have ∑

j∈J

∑
i∈Ij

〈Γji Λjx, Γji Λjx〉≤
∑
j∈J

Dj〈Λjx, Λjx〉

≤ DDΛ〈x,x〉.

On the other hand ∑
j∈J

∑
i∈Ij

〈Γji Λjx, Γji Λjx〉≥
∑
j∈J

CjΛjx, Λjx〉

≥ CCΛ〈K∗x,K∗x〉.

Assume now that {Γji Λj}j∈J,i∈Ij is a K −g−frame for H with bounds A,B. For each x ∈ H

Cj〈Λjx, Λjx〉≤
∑
i∈Ij

〈Γji Λjx, Γji Λjx〉≤ Dj〈Λjx, Λjx〉,

then ∑
j∈J
〈Λjx, Λjx〉≤

∑
j∈J

1

Cj

∑
i∈Ij

〈Γji Λjx, Γji Λjx〉

≤
1

c

∑
j∈J

∑
i∈Ij

〈Γji Λjx, Γji Λjx〉

≤
B

C
〈x,x〉.

On the other hand ∑
j∈J
〈Λjx, Λjx〉≥

∑
j∈J

1

Dj

∑
i∈Ij

〈Γji Λjx, Γji Λjx〉

≥
1

D

∑
j∈J

∑
i∈Ij

〈Γji Λjx, Γji Λjx〉

≥
A

D
〈K∗x,K∗x〉.

Therefore {Λj}j∈J being a K −g−frame for H is equivalent to {Γji Λj}j∈J,i∈Ij being a K −g−frame
for H, it remains only to prove the duality, then for each x ∈ H, we have∑

j∈J
Λ∗j Λ

′
jx =

∑
j∈J

Λ∗j
(∑
i∈Ij

Γ∗ji Γ
′
ji Λ

′
jx
)

=
∑
j∈J

∑
i∈Ij

Λ∗j Γ
∗
ji Γ
′
ji Λ

′
jx

=
∑
j∈J

∑
i∈Ij

(Γji Λj)
∗Γ
′
ji Λ

′
jx,



6 Int. J. Anal. Appl. (2022), 20:24

so {Λ
′
j}j∈J is a dual K−g−bessel sequence of {Λj}j∈J if and only if {Γ

′
ji Λ

′
j}j∈J,i∈Ij is a dual K−g−bessel

sequence of {Γji Λj}j∈J,i∈Ij . �

Theorem 2.3. Let {Λj}j∈J be a Parseval K−g−frame for H where K ∈ End∗A(H) has closed range.
Then {Λj(K+)∗}j∈J is a dual K −g−bessel sequence of {Λj}j∈J.

Proof. For each x ∈ H, we have∑
j∈J
〈Λj(K+)∗x, Λj(K+)∗x〉 = 〈K∗(K+)∗x,K∗(K+)∗x〉

≤ ‖K‖2‖K+‖2〈x,x〉,

then {Λj(K+)∗}j∈J is g−bessel sequence for H. And we have for each g ∈ range(K∗), g =
K∗(K∗)+g = K∗(K+)∗g, so

Kg = KK∗(K+)∗g =
∑
j∈J

Λ∗j Λj(K
+)∗g.

If h ∈ (range(K∗))⊥ = Ker(K) we obtain h ∈ Ker((K+)∗) implying that
∑
j∈J Λ

∗
j Λj(K

+)∗h = 0 =

Kh, altogether we have Kf =
∑
j∈J Λ

∗
j Λj(K

+)∗f for each f ∈ H. �

Theorem 2.4. Suppose that K ∈ End∗A(H) has closed range and that {Λj ∈ End
∗
A(H,Hj)}j∈J is a

Parseval K −g−frame for H. Then the following condition hold.

(1) For any dual K −g−bessel sequence {θj}j∈J of {Λj}j∈J we have T∗Λ̃TΛ̃ = T
∗
Λ̃
Tθ, where TΛ̃ is

the analysis operator of {Λj(K+)∗}j∈J.
(2) If {Γj ∈ End∗A(H,Hj)}j∈J is also a Parseval K − g−frame for H such that T

∗
ΛTΓ = 0, then

{Λj}j∈J and {Γj}j∈J admit a common dual K−g−bessel sequence {Λj(K+)∗ + Γj(K+)∗}j∈J.

Proof. Assume that {θj}j∈J is a dual K −g−bessel sequence of {Λj}j∈J. So

T∗Λ (TΛ̃x −Tθx) =
∑
j∈J

Λ∗j Λj(K
+)∗x −

∑
j∈J

Λ∗j θjx

= Kx −Kx = 0,

then,

〈T∗
Λ̃

(T
Λ̃
−Tθ)x,y〉 = 〈K+T∗Λ (TΛ̃ −Tθ)x,y〉 = 0,

hence, T∗
Λ̃

(T
Λ̃
−Tθ)x = 0, so T∗Λ̃TΛ̃ = T

∗
Λ̃
Tθ. (2) Since T∗ΛTΓ = 0, then∑

j∈J
Λ∗j (Λj(K

+)∗ + Γj(K
+)∗)x =

∑
j∈J

Λ∗j Λj(K
+)∗x

= Kx

=
∑
j∈J

Γ∗j Γj(K
+)∗x

=
∑
j∈J

Γ∗j (Λj(K
+)∗ + Γj(K

+)∗)x



Int. J. Anal. Appl. (2022), 20:24 7

�

Theorem 2.5. Let {Λj ∈ End∗A(H,Hj); j ∈ J} be a K −g−frame for H with bound A and B. Then
{Λj}j∈J is a g−frame for H if K∗ is bounded below.

Proof. Since K∗ is bounded below, then there exists C > 0 such that

〈K∗x,K∗x〉≤ C〈x,x〉,∀x ∈ H.

And we have

A〈K∗x,K∗x〉≤
∑
j∈J
〈Λjx, Λjx〉≤ B〈x,x〉,∀x ∈ H.

So,

AC〈x,x〉≤ A〈K∗x,K∗x〉≤
∑
j∈J
〈Λjx, Λjx〉≤ B〈x,x〉,∀x ∈ H.

Hence, {Λj}j∈J is a g−frame for H. �

Theorem 2.6. If {Λj ∈ End∗A(H,Hj); j ∈ J} is a tight K−g−frame with bounded A, then {Λj}j∈J is
a tight g−frame with bounded B if and only if the right inverse of the operator K is A

B
K∗.

Proof. Now if {Λj}j∈J is a tight g−frame with bounded B, then∑
j∈J
〈Λjx, Λjx〉 = B〈x,x〉,∀x ∈ H.

Since {Λj}j∈J is a tight K −g−frame with bounded A then,

A〈K∗x,K∗x〉 = B〈x,x〉,∀x ∈ H,

hence,

〈KK∗x,x〉 = 〈
B

A
x,x〉,

therefore,

KK∗ =
B

A
IH,

so, A
B
K∗ is the right inverse of the operator K.

Conversely, assume that A
B
K∗ is the right inverse of the operator K, then

KK∗ =
B

A
IH

so,

A〈KK∗x,x〉 = B〈x,x〉,

hence,

A〈K∗x,K∗x〉 = B〈x,x〉,

since, {Λj}j∈J is tight K −g−frame with bound A, then

B〈x,x〉 =
∑
j∈J
〈Λjx, Λjx〉,∀x ∈ H.



8 Int. J. Anal. Appl. (2022), 20:24

�

Theorem 2.7. Let I ⊂ J be given. Suppose that {Λj ∈ End∗A(H,Hj), j ∈ J} is a K −g−frame with
bounds A,B and K −g−frame operator SΛ,J. Then the following statements are equivalent:

(1) Irange(K) −S−1Λ,JSΛ,I is boundedly invertible on range(K),
(2) The sequence {Λj}j∈J−I is a K − g−frame for H with lower K − g−frame bound

B−1

‖S−1
Λ,J
‖2‖K∗(Irange(K)−S

−1
Λ,J
SΛ,I)

−1‖2
.

Proof. Denote the frame operator of the K −g−frame {Λj}j∈J−I by SΛ,J−I. We have

SΛ,J−I = SΛ,J −SΛ,I = SΛ,J(Irange(K) −S
−1
Λ,JSΛ,I),

then, {Λj}j∈J−I is a K −g−frame if and only if Irange(K) −S−1Λ,JSΛ,I is boundedly invertible.
Suppose that Irange(K) −S−1Λ,JSΛ,I is invertible. for any x ∈ H

x = S−1Λ,JSΛ,Jx

= S−1Λ,J

(∑
j∈J

Λ∗j Λjx +
∑
j∈J−I

Λ∗j Λjx

)
= S−1Λ,JSΛ,Ix +

∑
j∈J−I

S−1Λ,JΛ
∗
j Λjx.

So,

(Irange(K) −S
−1
Λ,JSΛ,I)x =

∑
j∈J−I

S−1Λ,JΛ
∗
j Λjx.

Hence,

‖(Irange(K) −S
−1
Λ,JSΛ,I)x‖ = ‖

∑
j∈J−I

S−1Λ,JΛ
∗
j Λjx‖

= sup
‖y‖=1

∣∣∣∣
∣∣∣∣〈 ∑
j∈J−I

S−1Λ,JΛ
∗
j Λjx,y〉

∣∣∣∣
∣∣∣∣

= sup
‖y‖=1

∣∣∣∣
∣∣∣∣ ∑
j∈J−I

〈Λjx, ΛjS−1Λ,Jy〉
∣∣∣∣
∣∣∣∣

≤ sup
‖y‖=1

∣∣∣∣
∣∣∣∣ ∑
j∈J−I

〈Λjx, Λjx〉
∣∣∣∣
∣∣∣∣

1
2
∣∣∣∣
∣∣∣∣ ∑
j∈J−I

〈ΛjS−1Λ,Jy, ΛjS
−1
Λ,Jy〉

∣∣∣∣
∣∣∣∣

1
2

≤
√
B‖S−1Λ,J‖

∣∣∣∣
∣∣∣∣ ∑
j∈J−I

〈Λjx, Λjx〉
∣∣∣∣
∣∣∣∣

1
2

,

therefore,

‖(Irange(K) −S
−1
Λ,JSΛ,I)x‖≤

√
B‖S−1Λ,J‖

∣∣∣∣
∣∣∣∣ ∑
j∈J−I

〈Λjx, Λjx〉
∣∣∣∣
∣∣∣∣

1
2

,



Int. J. Anal. Appl. (2022), 20:24 9

then Irange(K) −S−1Λ,JSΛ,I is well defined. If Irange(K) −S
−1
Λ,J
SΛ,I is invertible on H, then for any x ∈ H

we have

‖K∗x‖≤‖K∗(Irange(K) −S
−1
Λ,JSΛ,I)

−1‖‖(Irange(K) −S
−1
Λ,JSΛ,I)x‖

hence,
B−1

‖S−1
Λ,J
‖2‖K∗(Irange(K) −S−1Λ,JSΛ,I)−1‖2

‖K∗x‖2 ≤‖
∣∣∣∣
∣∣∣∣ ∑
j∈J−I

〈Λjx, Λjx〉
∣∣∣∣
∣∣∣∣.

�

Definition 2.1. Consider two g−bessel sequences {Λj ∈ End∗A(H,Hj), j ∈ J} and {θj ∈
End∗A(H,Hj), j ∈ J}. The sequence {Λj}j∈J and {θj}j∈J are said to be approximately K − g−dual
frames if ‖Irange(K) −TΛT∗θ‖ < 1. In this case, we say that {θj}j∈J is an approximate K −g−dual of
{Λj}j∈J.

Theorem 2.8. If {θj}j∈J is an approximate K −g−dual of {Λj}j∈J, then {θj(TΛT∗θ )
−1}j∈J is a K −

g−dual of {Λj}j∈J.

Proof. it is easy to see that {θj(TΛT∗θ )
−1}j∈J is a g−bessel sequence and

x = (TΛT
∗
θ )(TΛT

∗
θ )
−1x

=

∞∑
j=0

Λ∗j θj(TΛT
∗
θ )
−1x

=

∞∑
j=0

Λ∗j

(
θj

∞∑
j=0

(Irange(K) −−TΛT
∗
θ )
nx

)
.

Then, θj(TΛT∗θ )
−1 = {θj

∑∞
j=0(Irange(K) −−TΛT

∗
θ )
n}j∈J is a K −g−dual of {Λj}j∈J. �

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

publication of this paper.

References

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https://doi.org/10.1090/S0002-9947-1952-0047179-6.

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https://doi.org/10.2307/2372552
https://doi.org/10.2307/2372552


10 Int. J. Anal. Appl. (2022), 20:24

[8] F.D. Nhari, R. Echarghaoui, M. Rossafi, K−g−Fusion Frames in Hilbert C∗−Modules, Int. J. Anal. Appl. 19 (6)
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[9] 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. https://doi.org/10.1007/s11785-022-01229-4.

[10] M. Rossafi, F.D. Nhari, Controlled K−g−Fusion Frames in Hilbert C∗−Modules, Int. J. Anal. Appl. 20 (2022), 1.
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[11] M. Rossafi, S. Kabbaj, ∗-K-Operator Frame for End∗A(H), Asian-Eur. J. Math. 13 (2020), 2050060. https:
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[12] M. Rossafi, S. Kabbaj, Operator Frame for End∗A(H), J. Linear Topol. Algebra, 8 (2019), 85-95.
[13] M. Rossafi, S. Kabbaj, ∗-K-g-Frames in Hilbert A-Modules, J. Linear Topol. Algebra, 7 (2018), 63-71.
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Math. 17 (2018), 17-25. https://doi.org/10.2478/aupcsm-2018-0002.

[15] M. Rossafi, S. Kabbaj, Generalized Frames for B(H,K), Iran. J. Math. Sci. Inf. accepted.

https://doi.org/10.28924/2291-8639-19-2021-836
https://doi.org/10.1007/s11785-022-01229-4
https://doi.org/10.28924/2291-8639-20-2022-1
https://doi.org/10.1142/S1793557120500606
https://doi.org/10.1142/S1793557120500606
https://doi.org/10.2478/aupcsm-2018-0002

	1. introduction
	2. Main Results
	References