International Journal of Analysis and Applications

Volume 17, Number 1 (2019), 1-13

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

DOI: 10.28924/2291-8639-17-2019-1

CONTROLLED ∗-G-FRAMES AND ∗-G-MULTIPLIERS IN HILBERT
PRO-C∗-MODULES

ZAHRA AHMADI MOOSAVI1,∗ AND AKBAR NAZARI2

1Department of Mathematics Faculty of Mathematics and computer Shahid Bahonar University of Kerman,

76169-14111, Kerman, Iran

2Department of Mathematics Faculty of Mathematics and computer Shahid Bahonar University of Kerman,

76169-14111, Kerman, Iran

∗Corresponding author: nazari@uk.ac.ir

Abstract. A generalization of multiplier, controlled g-frames and g-Bessel sequences to ∗-g-frames and

∗-g-Bessel sequences in Hilbert pro-C∗-modules is presented. It is demonstrated that controlled ∗-g-frames

are equivalent to ∗-g-frames in Hilbert pro-C∗-modules.

1. Introduction

Frame theory is an application of harmonic analysis. This theory has been rapidly generalized to Hilbert

spaces and Hilbert C∗-modules. In 2005, Sun [22] introduced the notion of g-frames as a generalization of

frames for bounded operators on Hilbert spaces. Frank-Larson [5] have extended the theory for elements of

C∗-algebras and (finitely or countably generated) Hilbert C∗-modules have been considered in [1].

It is well known that Hilbert C∗-modules are a generalization of Hilbert spaces where the inner product

takes values in a C∗-algebra rather than in the field of complex numbers. The theory of Hilbert C∗-modules

Received 2018-04-13; accepted 2018-06-22; published 2019-01-04.

2010 Mathematics Subject Classification. 42C15, 46L08.

Key words and phrases. Hilbert pro-C∗-modules; ∗-g-frames; ∗-g-Bessel sequences; controlled ∗-g-frames; (C, C′)-controlled

∗-g-frames.
c©2019 Authors retain the copyrights

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

1

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-1


Int. J. Anal. Appl. 17 (1) (2019) 2

has applications in the study of locally compact quantum groups, complete maps between C∗-algebras, non-

commutative geometry and KK-theory. Not all properties of Hilbert spaces hold in Hilbert C∗-modules. For

instance, the Riesz representation theorem for continuous linear functionals on Hilbert spaces can not be

extended to Hilbert C∗-modules [23] and there exist closed subspaces in Hilbert C∗-modules that have no

orthogonal complement [16]. Moreover, as known, every bounded operator on a Hilbert space has an adjoint

whereas there are bounded operators on Hilbert C∗-modules which do not have this property [17]. So, it is

to be expected that frames and ∗-frames in Hilbert C∗-modules are more complicated than those in Hilbert

spaces. The properties of g-frames for Hilbert C∗-modules have been widely investigated in the literature (

see [1, 5, 12, 25], and the references therein).

The paper is organized as follows.In the next section, we give a brief survey of the fundamental definitions

and notations of Hilbert pro-C∗-modules.

Section 3 is devoted to investigating ∗-g-frames with A-valued bounds and analyzing their elementary

properties. In Section 4 we define the concept of controlled ∗-g-frames and we show that a controlled ∗-g-

frame is equivalent to a ∗-g-frame in Hilbert pro-C∗-modules. Finally, in section 5 we define multipliers of

controlled ∗-g-frame operators in Hilbert pro-C∗-modules.

2. Preliminaries

In this section, we recall some of the basic definitions and properties of pro-C∗-algebras and Hilbert

modules over them [7, 15, 18].

A pro-C∗-algebra is a complete Hausdorff complex topological ∗-algebra A whose topology is determined

by its continuous C∗-seminorms in the sense that a net {aλ} converges to 0 iff ρ(aλ) → 0 for any continuous

C∗-seminorm ρ on A and we have:

(1) ρ(ab) ≤ ρ(a)ρ(b);

(2) ρ(a∗a) = ρ(a)2;

for all C∗-seminorms ρ on A and a,b ∈A.

If the topology of pro-C∗-algebra is determined by only countably many C∗-seminorms, then it is called

a σ-C∗-algebra.

Let A be a unital pro-C∗-algebra with unit 1A and let a ∈ A . Then spectrum sp(a) of a ∈ A is the

set {λ ∈ C : λ1A − a is not invertible}. If A is not unital, then the spectrum is taken with respect to its

unitization Ã.

If A+ denotes the set of all positive elements of A, then A+ is a closed convex C∗-seminorms on A. We

denote by S(A), the set of all continuous C∗-seminorms on A.

Example 2.1. Every C∗-algebra is a pro-C∗-algebra.



Int. J. Anal. Appl. 17 (1) (2019) 3

Example 2.2. A sub-closed ∗-algebra of a pro-C∗-algebra is a pro-C∗-algebra.

Proposition 2.1 ( [6]). Let A be a unital pro-C∗-algebra with an identity 1A.Then for any ρ ∈ S(A), we

have:

(1) ρ(a) = ρ(a∗) for all a ∈ A;

(2) ρ(1A) = 1;

(3) If a,b ∈A+ and a ≤ b, then ρ(a) ≤ ρ(b);

(4) If 1A ≤ b, then b is invertible and b−1 ≤ 1A;

(5) If a,b ∈A+ are invertible and 0 ≤ a ≤ b, then 0 ≤ b−1 ≤ a−1;

(6) If a,b,c ∈A and a ≤ b then c∗ac ≤ c∗bc;

(7) If a,b ∈A+ and a2 ≤ b2, then 0 ≤ a ≤ b.

Definition 2.1. A pre-Hilbert module over pro-C∗-algebra A, is a complex vector space E which is also

a left A-module compatible with the complex algebra structure, equipped with an A-valued inner product

〈., .〉 : E ×E →A which is C-and A-linear in its first variable and satisfies the following conditions:

(1) 〈x,y〉∗ = 〈y,x〉;

(2) 〈x,x〉≥ 0;

(3) 〈x,x〉 = 0 iff x = 0;

for every x,y ∈ E. We say E is a Hilbert A-module (or Hilbert pro-C∗-module overA) If E is complete with

respect to the topology determined by the family of seminorms

ρE(x) =
√
ρ(〈x,x〉) x ∈ E,ρ ∈ S(A).

Let E be a pre-Hilbert A-module.By [6], for ρ ∈ S(A) and for all x,y ∈ E, the following Cauchy-

Bunyakovskii inequality holds:

ρ(〈x,y〉)2 ≤ ρ(〈x,x〉)ρ(〈y,y〉).

Consequently, for each ρ ∈ S(A), we have:

ρE(ax) ≤ ρ(a)ρ(x), a ∈A,x ∈ E.

Let A be a pro-C∗-algebra and E and F be two Hilbert A-modules. An A-module map T : E → F is

said to bounded if for each ρ ∈ S(A), there is Cρ > 0 such that :

ρF (Tx) ≤ Cρ. ρE(x) (x ∈ E),

where ρE, respectively ρF , are continuous seminorms on E, respectively F. A bounded A-module map from

E to F is called an operators from E to F . We denote the set of all operators from E to F by HomA(E,F),

and we set HomA(E,F) = EndA(E)



Int. J. Anal. Appl. 17 (1) (2019) 4

Proposition 2.2. Let T∗ ∈ HomA(E,F). We say T is adjointable if there exists an operator T∗ ∈ T ∈

HomA(F,E) such that:

〈Tx,y〉 = 〈x,T∗y〉

holds for all x ∈ E,y ∈ F.

We denote by Hom∗A(E,F), the set of all adjointable operator from E to F and End
∗
A(E) = Hom

∗
A(E,E)

Proposition 2.3 ( [6]). Let T : E → F and T∗ : F → E be two maps such that the equality

〈Tx,y〉 = 〈x,T∗y〉

holds for all x ∈ E, y ∈ F.Then T ∈ Hom∗A(E,F).

It is easy to see that for any ρ ∈ S(A), the map defined by:

ρ̂E,F (T) = sup{ρF (T(x) : x ∈ E, ρE(x) ≤ 1}, T ∈ HomA(E,F),

is a seminorm on HomA(E,F).

Definition 2.2. Let E and F be two Hilbert modules over pro-C∗-algebra A. Then the operator T : E → F

is called uniformly bounded (below), if there exists C > 0 such that:

ρF (Tx) ≤ C ρE(x). (2.1)

(C ρE(x) ≤ ρF (Tx)) (2.2)

The number C in (2.1) is called an upper bound for T and we set :

‖T‖∞ = inf{C : C is an upper bound for T}.

Clearly, in this case we have:

ρ̂(T) ≤‖T‖∞, ∀ρ ∈ S(A).

Let T be an invertible element in End∗A(E) such that both are uniformly bounded. Then by [2, Proposition

3.2], for each x ∈ E we have the inequality

‖T−1‖−2∞ 〈x,x〉≤ 〈Tx,Tx〉≤ ‖T‖
2
∞〈x,x〉. (2.3)

The following proposition will be used in the next section.

Proposition 2.4 ( [6]). Let T be an uniformly bounded below operator in HomA(E,F). then T is closed(range)

and injective.



Int. J. Anal. Appl. 17 (1) (2019) 5

3. ∗-G-frames in Hilbert pro-C∗-modules

Throughout this section A is a pro-C∗-algebra, U and V are two Hilbert A-modules. also {Vj}j∈J is a

countable sequence of closed submodules of V .

Definition 3.1. A sequence Λ = {Λj ∈ Hom∗A(U,Vj)}j∈J is called a ∗- g-frame for U with respect to

{Vj}j∈J if

C〈f,f〉C∗ ≤
∑
j∈J〈Λjf, Λjf〉≤ D〈f,f〉D

∗

for all f ∈ U and strictly nonzero elements C,D ∈A.

The number C and D are called ∗-g-frame bounds for Λ. The ∗-g-frame is called tight if C = D and a

Parseval if C = D = 1. If in the above we only have the upper bound, then Λ is called a ∗-g-Bessel sequence.

Also if for each j ∈ J,Vj = V , we call Λ a ∗-g-frame for U with respect to V .

We mentioned that the set of all g-frames in Hilbert pro-C∗-modules are a subset of the family of ∗-

g-frames. To illustrate this, let Λ = {Λj}j∈J be a g-frame for U with respect to {Vj}j∈J. Note that for

f ∈ U,

(
√
C)1A〈f,f〉(

√
C)1A ≤

∑
j∈J〈Λjf, Λjf〉(

√
D)1A〈f,f〉(

√
D)1A

Therefore, every g-frame for U with real bounds C amd D is a ∗-g-frame for U with A-valued ∗-g-frame

bounds (
√
C)1A and (

√
D)1A.

Example 3.1. Let `2(A) be the set of all sequences (an)n∈N of elements of a pro-C∗-algebra A such that

the series
∑
i∈N aia

∗
i is convergent in A. Then, by [2, Example 3.2], `

2(A) is a Hilbert module over A with

respect to pointwise operations and inner product defined by:

〈(ai)i∈N, (bi)i∈N〉 =
∑
i∈N

aib
∗
i .

Let a = (ai)i∈N and b = (bi)i∈N in `
2(A). We define ab = {aibi}i∈N and ρ(a) =

√
ρ(〈a,a〉) and a∗ :=

{ai}i∈N and 〈a,b〉 = ab∗ =
∑
i∈N aib

∗
i .

Now, let j ∈ J := N and define fj ∈ `2(A) by fj = {f
j
i }i∈N such that

f
j
i =




1

i
1A i = j;

0 i 6= j, ∀j ∈ N.

Set Λj : `
2(A) →A by Λfj (U) = 〈U,fj〉 for any U ∈ `2(A) . We see that∑

j∈J〈Λfj (U), Λfj (U)〉≤ 〈U,U〉.

Thus{Λj}j∈J is a ∗-g-Bessel sequence .



Int. J. Anal. Appl. 17 (1) (2019) 6

Definition 3.2. Let Λ = {Λj ∈ End∗A(U,Vj)}j∈J be a ∗-g-frame for U with respect to {Vj}j∈J with bounds

C and D. We define the corresponding ∗-g-frame transform as follows:

TΛ : U →
⊕

j∈J Vj , TΛf = {Λjf : j ∈ J}, for all f ∈ U.

Since Λ is a ∗-g-frame, hence for each f ∈ U we have:

C 〈f,f〉 C∗ ≤
∑
j∈J〈Λjf, Λjf〉≤ D 〈f,f〉 D

∗,

So TΛ is well-defined. Also for any ρ ∈ S(A) and f ∈ U the following inequality is obtained:

ρ(C)2 ρU (f) ≤ ρ⊕
j
Vj

(TΛf) ≤ ρ(D)2 ρU (f).

From the above, it follows that the ∗-g-frame transform is an uniformly bounded below operator in EndA(U,
⊕

j∈J Vj).

Thus by Proposition 2.4, TΛ is closed and injective.

Now, we define the synthesis operator for ∗-g-frame Λ as follows:

T∗Λ :
⊕
j∈j

Vj → U, T∗Λ({yj}j) =
∑
j∈J

Λ∗j (yj), (3.1)

where Λ∗j is the adjoint operator of Λj.

Proposition 3.1. The synthesis operator defined by (3.1) is well-defined, uniformly bounded and the adjoint

of the transform operator.

Proof. Since Λ = {Λj : j ∈ J} is a ∗-g-frame for U with respect to {Vj}j∈J, there exist C,D ∈A such that

for any f ∈ U,

C 〈f,f〉 C∗ ≤
∑
j∈J〈Λjf, Λjf〉≤ D 〈f,f〉 D

∗.

Let I be an arbitrary finite subset of J. Using the Cauchy-Bunyakovskii inequality and [24, Lemma 2.2], for

any ρ ∈ S(A) and (yj)j ∈
⊕

j∈J Vj we have:

ρ(
∑
j∈I

Λ∗j (yj)) = sup{ρ〈
∑
j∈I

Λ∗j (yj),f〉 : f ∈ U , ρ(f) ≤ 1}

= sup{ρ(
∑
j∈I
〈yj, Λjf〉) : f ∈ U , ρ(f) ≤ 1}

≤ sup
ρ(f)≤1


ρ(∑

j∈I
〈yj,yj〉)


1/2


ρ(∑

j∈I
〈Λjf, Λjf〉)


1/2

≤ sup
ρ(f)≤1

ρ(DD∗)1/2ρ(f)(ρ
∑
j∈I
〈yj,yj〉)1/2

≤


ρ(D) (ρ∑

j∈I
〈yj,yj〉)1/2


 .



Int. J. Anal. Appl. 17 (1) (2019) 7

Now, since the series
∑
j∈J〈yj,yj〉 converges in A, the above inequality shows that

∑
j∈J Λ

∗
j (yj) is convergent.

Hence T∗Λ is well-defined. On the other hand, for any f ∈ U and (yj)j ∈
⊕

j∈J Vj, we have:

〈TΛ(f), (yj)j〉 = 〈(Λjf)j, (yj)j〉

=
∑
j∈J
〈Λjf,yj〉

=
∑
j∈J
〈f, Λ∗jyj〉

= 〈f,
∑
j∈J

Λ∗jyj〉

= 〈f,T∗Λ(yj)j∈J〉.

Therefore by Proposition 2.2 it follows that the synthesis operator is the adjoint of the transform operator.

Also, for any ρ ∈ S(A) we have:

ρU (T
∗
Λ(y)) ≤ ρ(D) ρ⊕j∈J Vj (y), y = (yj)j ∈

⊕
j∈J Vj.

Hence the synthesis operator is uniformly bounded. �

Let Λ = {Λj , j ∈ J} be a ∗-g-frame for U with repect to {Vj}j∈J. Define the corresponding ∗-g-frame

operator SΛ as follows:

SΛ = T
∗
ΛTΛ : U → U SΛ(f) =

∑
j∈J Λ

∗
j Λjf.

Since SΛ is a combination of two bounded operators, it is a bounded operator.

Theorem 3.1. Let Λ = {Λj}j∈J be a ∗-g-frame for U with respect to {Vj}j∈J and with bounds C,D. Then

SΛ is an invertible positive operator. Also it is a self-adjoint operator such that:

CIUC
∗ ≤ SΛ ≤ DIUD∗. (3.2)

Here IU is the identity function on U.

Proof. According to the definition of the transform operator, for any f ∈ U we can write:

〈TΛ(f),TΛ(f)〉 = 〈{Λjf}j∈J,{Λjf}j∈J〉 =
∑
j∈J〈Λjf, Λjf〉.

Since Λ is a ∗-g-frame for U with bounds C and D, for each f ∈ U it follows that

C〈f,f〉C∗ ≤〈TΛ(f),TΛ(f)〉≤ D〈f,f〉D∗.

On the other hand,

〈SΛ(f),f〉 = 〈T∗ΛTΛ(f),f〉 = 〈TΛ(f),TΛ(f)〉 = 〈f,T
∗
ΛTΛ(f)〉 = 〈f,SΛ(f)〉.

Consequently, SΛ is a self-adjoint operator. Also, for any f ∈ U, we obtain



Int. J. Anal. Appl. 17 (1) (2019) 8

C〈f,f〉C∗ ≤〈SΛ(f),f〉≤ D〈f,f〉D∗.

It follows that ∗-g-frame operator is positive and (3.2) also holds. Moreover, since SΛ is one-to-one it follows

that SΛ is invertible. �

According to (3.2) and Proposition 2.1 we have the following Lemma

Lemma 3.1.

D−1IU (D
−1)∗ ≤ S−1Λ ≤ C

−1IU (C
−1)∗.

Hence the ∗-g-frame operator and its inverse belong to End∗A(U).

Theorem 3.2. Let {Λj ∈ End∗A(U,Vj)}j∈J and
∑
j∈J〈Λjf, Λjf〉 converge in the semi-norm for f ∈ U.

Then Λ = {Λj}j∈J is a ∗-g-frame for U with respect to {Vj}j∈J if and only if there are two strictly nonzero

elements C,D ∈A such that for every f ∈ U,

ρ(C−1)−1 ρ(〈f,f〉)ρ(C∗−1)−1 ≤ ρ(
∑
j∈J
〈Λjf, Λjf〉)

≤ ρ(D) ρ(〈f,f〉)ρ(D∗). (3.3)

Proof. If {Λj ∈ End∗A(U,Vj)}j∈J is a ∗-g-frame for U with respect to {Vj}j∈J, then

(〈f,f〉) ≤ C−1(
∑
j∈J
〈Λjf, Λjf〉)(C∗)−1)

and

(
∑
j∈J
〈Λjf, Λjf〉) ≤ D〈f,f〉D∗.

Therefore, by Proposition 2.1,

ρ(C−1)−1 ρ(〈f,f〉)ρ(C∗−1)−1 ≤ ρ(
∑
j∈J
〈Λjf, Λjf〉)

≤ ρ(D) ρ(〈f,f〉)ρ(D∗). (3.4)

For the converse, let (3.3) hold. Then we define a linear operator as follows:

M : U →
⊕
j∈J

Vj, M(f) = {Λjf}j∈J, ∀f ∈ U,

〈Mf,Mf〉 =
∑
j∈J
〈Λjf, Λjf〉, ∀f ∈ U.

Hence, by (3.3), we have

ρU (M(f)) ≤ ρ(D)
1
2 ρU (f) ρ(D

∗)
1
2 .



Int. J. Anal. Appl. 17 (1) (2019) 9

This shows that M is uniformly bounded. We write M∗M = K. Then 〈M(f),M(f)〉 = 〈M∗M(f),f〉 =

〈K(f),f〉. Therefore, K is positive. As, K∗ = (M∗M),K is self-adjoint. On the other hand,

〈K
1
2 f,K

1
2 f〉 = 〈Kf,f〉 =

∑
j∈J
〈Λjf, Λjf〉.

Now, according to Proposition 2.4 and (3.3), K
1
2 is invertible and uniformly bounded; therefore, by [2,

Proposition 3.2], we have:

‖K−
1
2‖−1∞ 〈f,f〉‖K

−1
2‖−1∞

∗
≤〈K

1
2 (f),K

1
2 (f)〉≤ ‖K

1
2‖∞〈f,f〉‖K

1
2‖∞

Hence {Λj}j∈J is a ∗-g-frame. �

4. Controlled ∗-G-frames in Hilbert pro-C∗-modules

In this section, we define the concept of multipliers for ∗-g-Bessel sequences and we show that controlled

∗-g-frames are equivalent to ∗-g-frames.

Let A be a pro-C∗-algebra, U and V be two Hilbert A-modules. also, let {Vj}j∈J be a countable sequence

of closed submodules of V , L(U,V ) and L(U) the collection of all bounded linear operators from U into V

and U respectively. gl(U) the set of all bounded operators with a bounded inverse and gl+(U) be the set of

positive operators in gl(U).

Proposition 4.1. Let Λ = {Λj ∈ L(U,Vj) : j ∈ J} and θ = {θj ∈ L(U,Vj) : j ∈ J} be ∗-g-Bessel sequences

with bounds BΛ and Bθ. If for m = {mj}j ⊆ `∞(R), the operator

M = Mm,Λ,θ : U → U

M(f) =
∑
j

mjΛ
∗
jθjf, (4.1)

is well-defined, then M is called the ∗-g-multiplier of Λ,θ and m.

Proof. Let I be an arbitrary finite subset of J. Using the Cauchy-Bunyakovskii inequality and [24, Lemma

2.2], for any ρ ∈ S(A) and f ∈ U we have:

ρ(
∑
j∈I

mjΛ
∗
jθjf) = sup{ρ〈

∑
j∈I

mjΛ
∗
jθjf,g〉 : g ∈ U , ρ(g) ≤ 1}

= sup{ρ(
∑
j∈I
〈mjθjf, Λjg〉) : g ∈ U , ρ(g) ≤ 1}

≤ sup
ρ(g)≤1


ρ(∑

j∈I
〈mjθjf,mjθjf〉)


1/2


ρ(∑

j∈I
〈Λjg, Λjg〉)


1/2 .



Int. J. Anal. Appl. 17 (1) (2019) 10

Since ∑
j

〈mjθjf,mjθjf〉 =
∑
j

mj〈θjf,θjf〉m∗j

=
∑
j

(ρ(mj))
2〈θjf,θjf〉

≤ ‖m‖2∞Bθ〈f,f〉B
∗
θ,

so by Proposition 2.1 we have:

ρ(
∑
j〈mjθjf,mjθjf〉) ≤‖m‖

2
∞(ρ(f))

2ρ(Bθ)
2.

Hence we have:

ρ(
∑
j∈I

mjΛ
∗
jθjf) ≤‖m‖∞ ρ(f) ρ(Bθ) ρ(BΛ)

�

Definition 4.1. Let C,C′ ∈ gl+(U). The family Λ = {Λj ∈ L(U,Vj) : j ∈ J} is called a (C,C′)-controlled

∗-g-frame for U with respect to {Vj}j∈J, if Λ is a ∗-g-Bessel sequence and

A〈f,f〉A∗ ≤
∑
j∈J
〈ΛjCf, ΛjC′f〉≤ B〈f,f〉B∗, (4.2)

for all f ∈ U and strictly nonzero elements A,B ∈A.

A,B are called controlled ∗-g-frame bounds. If C′ = I, we call Λ = {Λj}j a C-controlled ∗-g-frame for

U with bounds A,B. If only the second part of the above inequality holds, it is called a (C,C′)-controlled

∗-g-Bessel sequence with bound B.

Lemma 4.1 ( [2]). Let X be a Hilbert module over C∗-algebra B, S ≥ 0, i.e. this element is positive in

C∗-algebra L(U). Then for each x ∈ X,

〈Sx,x〉≤ ‖S‖〈x,x〉.

Proposition 4.2. Let C ∈ gl+(H). The family

Λ = {Λj ∈ L(U,Vj) : j ∈ J}

is a ∗-g-frame if and only if Λ is a C2- controlled ∗-g-frame.

Proof. Let Λ be a C2- controlled ∗-g-frame with bounds A,B. Then

A〈f,f〉A∗ ≤
∑
j∈J
〈ΛjCf, ΛjCf〉≤ B〈f,f〉B∗, for f ∈ U.

A〈f,f〉A∗ = A〈CC−1f,CC−1f〉A∗ ≤ A‖C‖2〈C−1f,C−1f〉A∗ ≤‖C‖2
∑
j∈J
〈ΛjCC−1f,CC−1f〉.



Int. J. Anal. Appl. 17 (1) (2019) 11

Hence

A‖C‖−1〈f,f〉A∗‖C‖−1 ≤
∑
j∈J
〈Λjf, Λjf〉.

On the other hand for every f ∈ U

∑
j∈J
〈Λjf, Λjf〉 =

∑
j∈J
〈ΛjCC−1f,CC−1f〉

≤ B〈C−1f,C−1f〉B∗

≤ B‖C−1‖2〈f,f〉B∗.

These inequalities yield that Λ is a ∗-g-frame with bounds A‖C−1‖,B‖C−1‖. Conversely assume that Λ is

a ∗-g-frame with bounds A′,B′. Then for all f ∈ U,

A′〈f,f〉A′
∗
≤

∑
j∈J
〈Λjf, Λjf〉≤ B′〈f,f〉B′

∗
.

So for f ∈ U, ∑
j∈J
〈ΛjCf, ΛjCf〉≤ B′〈Cf,Cf〉B′

∗
≤ B′‖C‖2B′

∗
.

For the lower bound, since Λ is ∗-g-frame for any f ∈ U,

A′〈f,f〉A′
∗

= A′〈C−1Cf,C−1Cf〉A′
∗

≤ A′‖C−1‖2〈Cf,Cf〉A′
∗

≤‖C−1‖2
∑
j∈J
〈ΛjCf, ΛjCf〉.

Therefor Λ is a C2-controlled ∗-g-frame with bounds A′‖C−1‖,B′‖C−1‖ �

5. Multipliers of controlled ∗-G-frames in Hilbert pro-C∗-modules

In this section, we define the multiplier of a controlled ∗-g-frame for C-controlled ∗-g-frames in Hilbert

pro-C∗-modules. The definition of general case (C,C′)-controlled ∗-g-frames is similar.

Lemma 5.1. Let C,C′ ∈ gl+(U) and Λ = {Λj ∈ L(U,Vj) : j ∈ J},θ = {θj ∈ L(U,Vj) : j ∈ J} be C′2 and

C2-controlled ∗-g-Bessel sequences for U, respectively. Let m = `∞ . Then

Mm,C,θ,Λ,C′ : U → U,

defined by

Mm,C,θ,Λ,C′f :=
∑
j∈J

mjCθ
∗
j ΛjC

′f,

is a well-defined bounded operator.



Int. J. Anal. Appl. 17 (1) (2019) 12

Proof. Let Λ = {Λj ∈ L(U,Vj) : j ∈ J},θ = {θj ∈ L(U,Vj) : j ∈ J} be C′2 and C2-controlled ∗-g-Bessel

sequences for U, with bounds B,B′, respectively. For any f,g ∈ U and finite subset I ⊆ J,

ρ(
∑
j∈I

mjCθ
∗
j ΛjC

′f) ≤ sup{ρ〈
∑
j∈I

mjCθ
∗
j ΛjC

′f,g〉 : g ∈ U , ρ(g) ≤ 1}

= sup{ρ(
∑
j∈I
〈mjΛjC′f,θjC∗g〉) : g ∈ U , ρ(g) ≤ 1}

≤ sup
ρ(g)≤1


ρ(∑

j∈I
〈mjΛjC′f,mjΛjC′f〉)


1/2


ρ(∑

j∈I
〈θjC∗g,θjC∗g〉)


1/2 ,

since

∑
j

〈mjΛjC′f,mjΛjC′f〉 =
∑
j

mj〈ΛjC′f, ΛjC′f〉m∗j

=
∑
j

(ρ(mj))
2〈ΛjC′f, ΛjC′f〉

≤ ‖m‖2∞B〈f,f〉B
∗.

So by Proposition 2.1 we have:

ρ(
∑
j

〈mjΛjC′f,mjΛjC′f〉) = ρ(
∑
j

mj〈ΛjC′f, ΛjC′f〉m∗j )

≤‖m‖2
∞

(ρ(f))2ρ(B)2.

Hence

ρ(
∑
j∈I

mjCθ
∗
j ΛjC

′f) ≤‖m‖∞ ρ(f) ρ(B) ρ(B)
′.

This shows that Mm,C,θ,Λ,C′ is well-defined and

ρ(Mm,C,θ,Λ,C′) ≤‖m‖∞ ρ(B) ρ(B)′. �

The above Lemma provides a motivation for the following definition.

Definition 5.1. Let C,C′ ∈ gl+(U) and Λ = {Λj ∈ L(U,Vj) : j ∈ J},θ = {θj ∈ L(U,Vj) : j ∈ J} be C′2

and C2-controlled ∗-g-Bessel sequences for U, respectively. Let m = `∞ . The operator

Mm,C,θ,Λ,C′ : U → U,

defined by

Mm,C,θ,Λ,C′f :=
∑
j∈J

mjCθ
∗
j ΛjC

′f,

is called (C,C′)-controlled multiplier operator with symbol m.



Int. J. Anal. Appl. 17 (1) (2019) 13

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	1. Introduction
	2.  Preliminaries
	3. -G-frames in Hilbert pro-C-modules
	4. Controlled -G-frames in Hilbert pro-C-modules 
	5. Multipliers of controlled  -G-frames in Hilbert pro-C-modules
	References