View of Controlled g-frames and their dual in Hilbert $C^{\ast}-$modules

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Abdelilah Karara¹, Mohamed Rossafi², Samir Kabbaj³

1Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco

2LaSMA Laboratory Department of Mathematics Faculty of Sciences, Dhar El Mahraz University Sidi Mohamed Ben Abdellah, P. O. Box 1796 Fez Atlas, Morocco

3Laboratory of Partial Differential Equations, Spectral Algebra and Geometry, Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco

E-mail: ¹[email protected], ²[email protected], ³[email protected]

Abstract.

In this paper we give some new results for controlled g-frames and controlled dual g-frames in HilbertC^∗-modules. First, we talk about controlled g-frame characterisation and find certain conditions that are equal to them. Then, we explain the purpose controlled dual g-frames and controlled dual g-frames operator and discuss some of their characteristics.

Keywords: g-frame, controlled g-frame,C^∗-algebras, HilbertC^∗-modules.

1. Introduction

The notion of frame is a recent active mathematical research topic, signal processing, com- puter science, etc. Frames for Hilbert spaces were first introduced in 1952 by Duffin and Schaefer [5] for study of nonharmonic Fourier series. Daubechies, Grossmann, and Meyer [4]

revived and developed them in1986, and popularized from then on.

In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C^∗-modules. For more details of frames in Hilbert C^∗-modules we refer to [8, 10, 13, 7]. Currently, the study of g-frames has yielded many results. Controlled frames have been introduced by Balazs et al.[1] to improve the numerical efficiency of iterative algorithms for inverting frame operator on abstract Hilbert spaces. Recently, Kouchi and Rahimi [11]

introduced Controlled frames in HilbertC^∗-modules.

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Controlled g-frames and their dual in HilbertC^∗−modules 11

In this paper we give the characterization of controlled g-frames and controlled dual g-frames in Hilbert C^∗-modules and also we characterize controlled g-frames and get some comparable conditions for them. In the end we present the notion of controlled dual frames in Hilbert C^∗-modules and give fundamental characterizations of these frames via operator machinery.

2. Preliminaries

Definition 2.1. [3]. LetAbe a Banach algebra, an involution is a mapx→x^∗ ofAinto itself such that for allxandy inAand all scalarsαthe following conditions hold:

(1) (x^∗)^∗=x.

(2) (xy)^∗=y^∗x^∗.

(3) (αx+y)^∗= ¯αx^∗+y^∗.

Definition 2.2. [3]. AC^∗-algebraAis a Banach algebra with involution such that : kx^∗xk=kxk²

for everyxinA.

Definition 2.3. [9]. Let Abe a unital C^∗-algebra and Hbe 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 anA-valued inner product h., .i:H × H → A, such that is sesquilinear, positive definite and respects the module action. In the other words,

(i) hx, xi ≥0 for allx∈ Handhx, xi= 0 if and only if x= 0.

(ii) hax+y, zi=ahx, zi+hy, zifor alla∈ Aandx, y, z∈ H.

(iii) hx, yi=hy, xi^∗ for allx, y∈ H.

Forx∈ H, we define||x||=||hx, xi||¹². IfHis complete with||.||is a norm onH, 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)¹²

Let Θ be a finite or countably index sets, N the set of natural numbers. For eachξ∈Θ, we also reserve the notation End^∗_A(H,Ki) for the collection of all adjointableA-linear maps from HtoKξ and End^∗_A(H,H) is denoted by End^∗_A(H). The set of all bounded linear operators on Hwith a bounded inverse is denoted byGL(H) andGL⁺(H) be the set of all positive bounded linear invertible operators onHwith bounded inverse. We also denote

M

ξ∈Θ

Kξ={α={αξ}:αξ ∈ Kξ and X

ξ∈Θ

hαξ, αξi is norm convergent in A } .

Letf ={fξ}_ξ∈Θ andg ={gξ}_ξ∈Θ, the inner product is defined byhf, gi=P

ξ∈Θhfξ, g_ξi, we haveL

ξ∈ΘKξ is a Hilbert A-module (see [12]).

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Definition 2.4. [14] A sequence{Υ_ξ ∈End^∗_A(H,K_i)}_i∈Jis called ag-frame inH, if there exist two constantsA, B >0 such that for everyf ∈ H,

(2.1) Ahf, fi ≤X

i∈J

hΥif,Υξfi ≤Bhf, fi.

The numbersAandBare called theg-frames bounds of{Υi ∈End^∗_A(H,Ki)}i. IfA=B=δ, theg-frame is calledδ-tight and ifA=B= 1, it is called a Parsevalg-frame. If only the right- hand inequality of (2.1) is satisfied,{Υξ}_ξ∈J is called ag-Bessel sequence forH.

Definition 2.5. Let{Υ_ξ ∈End^∗_A(H,K_ξ)}_ξ be ag-frames forHif f =X

ξ∈Θ

Φ^∗_ξΥξf forf ∈ H

{Φ_ξ}_ξ∈Θis called an alternate dualg-frame for{Υ_ξ}_ξ∈Θ. Furthermore,{Υ_ξ}_ξ∈Θis an alternate dualg-frame for{Φ_ξ}_ξ∈Θ, that is to say

f =X

ξ∈Θ

Υ^∗_ξΦξf forf ∈ H.

Definition 2.6. LetC1, C2∈GL⁺(H). A sequence of adjointable operators{Υξ}_ξ∈Θis called a (C1, C2)-controlled g-frame forH. If there exist two positive constantsA, B >0 such that

(2.2) Ahf, fi ≤X

ξ∈Θ

hΥ_ξC₁f,Υ_ξC₂fi ≤Bhf, fi. ∀f ∈ H

the numbersAand B are called the lower and upper frame bounds for (C1, C2)-controlled g-frame, respectively.

IfP

ξ∈ΘhΥξC1f,ΥξC2fi ≤Bhf, fifor allf ∈ H, then{Υξ}_ξ∈Θ is called a (C1, C2)-controlled g-Bessel sequence forHIfC2=IH, we call{Υξ}_ξ∈Θ aC1-controlledg-frame forH.

Lemma 2.7. [2] Let H be a HilbertA-module, All positive and bounded operator P :H → H has a unique positive and bounded square rootQ. We have

(1) P is self-adjoint⇒ Qis self-adjoint (2) P is invertible⇒ Qis invertible

Let{Υξ}_ξ∈J be (C1, C2)-controlled g-Bessel sequence with bound B, the operator : T_C₁_ΥC₂ :M

ξ∈Θ

K_ξ → H, T_C₁_ΥC₂

{f_ξ}_ξ∈Θ

=X

ξ∈Θ

(C₁C₂)¹²Υ^∗_ξf_ξ, ∀ {f_ξ}_ξ∈Θ∈M

ξ∈Θ

K_ξ

is called the synthesis operator of{Υξ}_ξ∈Θand T_C^∗₁_ΥC₂ :H →M

ξ∈Θ

Kξ, T_C^∗₁_ΥC₂f =n

Υ_ξ(C₂C₁)¹²fo

ξ∈Θ, ∀f ∈ H.

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T_C^∗

1ΥC₂ is called the analysis operator of{Υ_ξ}_ξ∈Θ.

WhenC1, C2 commute between them, and commute with the operator Υ^∗_ξΥξ for every ξ, the operator

SC₁ΥC₂ :H → H, SC₁ΥC₂f =X

ξ∈Θ

C2Υ^∗_ξΥξC1f, ∀f ∈ H

is called the frame operator of{Υξ}_ξ∈Θ. We have,SC₁ΥC₂ =C1SΥC2is positive and invertible, where SΥ is frame operator of g-frame {Υξ}_ξ∈Θ, and is positive, invertible, bounded, self- adjoint, andAI_H ≤SΥ≤BI_H.

For the above result one is referred to Hua and Huang [6]. therefore from now on we suppose thatC1, C2 commute between them, and commute with the operator Υ^∗_ξΥξ for everyξ.

3. Controlled g-frames in HilbertC^∗−modules

Theorem 3.1. [11] Let T : H → H be a linear operator. Then there exist two constants 0< C1≤C2<∞, such thatC1IH≤ T ≤C2IH if and only if T ∈GL⁺(H)

Lemma 3.2. Let C1, C2∈GL⁺(H). Then the following assertions are equivalent:

(1) {Υξ}_ξ∈Θ is a(C1, C2)-controlled g-frame forH with respect to{Kξ}_ξ∈Θ (2) {Υξ}_ξ∈Θ is a g-frame forHwith respect to {Kξ}_ξ∈Θ.

Proof. (1)⇒(2) Let{Υξ}_ξ∈Θ is a (C1, C2)-controlled g-frame for Hwith respect to {Kξ}_ξ∈Θ with boundsA, B, andh∈ H, we have

Ahh, hi=Ah(C₁C₂)¹²(C₁C₂)⁻¹²h,(C₁C₂)¹²(C₁C₂)⁻¹²hi

≤A

(C1C2)¹²

2

h(C1C2)⁻¹²h,(C1C2)⁻¹²hi

≤

(C1C2)¹²

2X

ξ∈Θ

D

ΥiC1(C1C2)⁻¹²h,ΥξC2(C1C2)⁻¹²hE

=

(C1C2)¹²

2* C2

X

ξ∈Θ

ΥξC1(C1C2)⁻¹²h,Υξ(C1C2)⁻¹²h +

=

(C1C2)¹²

2D

C2SΥC1(C1C2)⁻¹²h,(C1C2)⁻¹²hE

=

(C1C2)¹²

2

hSΥh, hi. So

A

(C₁C₂)¹²

2hh, hi ≤X

ξ∈Θ

hΥih,Υξhi, ∀h∈ H

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Forh∈ H,

hSΥh, hi=X

ξ∈Θ

hΥih,Υξhi=D

(C1C2)⁻¹²(C1C2)¹²SΥh, hE

=D

(C1C2)¹²SΥh,(C1C2)⁻¹²hE

=D

SΥ(C1C2)(C1C2)⁻¹²h,(C1C2)⁻¹²hE

=D

C₁S_ΥC₂(C₁C₂)⁻¹²h,(C₁C₂)⁻¹²hE

≤B

(C₁C₂)⁻¹²

2

hh, hi

Which implies that{Υξ}_ξ∈Θ is a g-frame for H with respect to{Kξ}_ξ∈Θ. (2)⇒(1) Suppose that{Υξ}_ξ∈Θ is a g-frame forHwith respect to{Kξ}_ξ∈Θ with boundsA₁, B₁. Then

hA1h, hi ≤ hSΥh, hi ≤ hB1h, hi for anyh∈ H.

SinceC1, C2∈GL⁺(H), by Lemma 3.1, there exist constantsr, r1, R, R1(0< r, r1, R, R1<∞) such that

rIH≤C1≤RIH, r1IH≤C2≤R1IH. UsinghC1SΥh, hi=hh, SΥC1hi=hh, C1SΥhi, we get

rA≤SΥC1=C1SΥ≤RB.

Identically, we have

rr1A≤C2SΥC1≤RR1B.

Thus

rr1Ahh, hi ≤X

ξ∈Θ

hΥξC1h,ΥξC2hi ≤RR1Bhh, hi, ∀h∈ H.

We conclude that{Υi ∈End^∗_A(H,Ki)}ξ∈J is a (C1, C2)-controlled g-frame forHwith respect

to{Kξ}_ξ∈Θ.

Lemma 3.3. Let C1, C2 ∈ GL⁺(H). Then {Υξ ∈ End^∗_A(H,Kξ)}ξ is a (C1, C2)-controlled g-frame for Hwith respect to{Kξ}_ξ∈Θ if and only if

Ahf, fi ≤X

ξ∈Θ

DΥ_ξ(C₂C₁)¹²f,Υ_ξ(C₂C₁)¹²fE

≤Bhf, fi, ∀f ∈ H,

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Proof. Letf ∈ H, we have X

ξ∈Θ

hΥ_ξC₁f,Υ_ξC₂fi=

* X

ξ∈Θ

C₂Υ^∗_ξΥ_ξC₁f, f +

=hC₂S_ΥC₁f, fi

=hC2C1SΥf, fi=D

(C2C1)¹²SΥ(C2C1)¹²f, fE

=

* X

ξ∈Θ

(C₂C₁)¹²Υ^∗_ξΥ_ξ(C₂C₁)¹²f, f +

=X

ξ∈Θ

D

Υξ(C2C1)¹²f,Υξ(C2C1)¹²fE

consequently, {Υξ:ξ∈Θ} is a (C1, C2)-controlled g-frame forH with respect to {Kξ}_ξ∈Θ is equivalent to

Ahf, fi ≤X

ξ∈Θ

D

Υξ(C2C1)¹²f,Υξ(C2C1)¹²fE

≤Bhf, fi, ∀f ∈ H,

Thus{Υξ :ξ∈Θ}is a

(C₂C₁)¹²,(C₂C₁)¹²

-controlled g-frame forHwith respect to{Kξ}_ξ∈Θ.

Lemma 3.4. Let C1, C2∈GL⁺(H). Then the following statements are equivalent:

(1) {Υξ}_ξ∈Θ is a(C1, C2)-controlledg-frame forHwith respect to{Kξ}_ξ∈Θ (2) {Υξ}_ξ∈Θ is aC2C1-controlledg-frame forHwith respect to{Kξ}_ξ∈Θ. Proof. LetC1, C2∈GL⁺(H),forf ∈ H, we have

X

ξ∈Θ

hΥξC1f,ΥξC2fi=

* X

ξ∈Θ

C2Υ^∗_ξΥξC1f, f +

=hC2SΥC1f, fi

=hC2C1SΥf, fi

=

* X

ξ∈Θ

(C2C1)Υ^∗_ξΥξf, f +

=X

ξ∈Θ

hΥ_ξ(C₂C₁)f,Υ_ξfi and we have

Ahf, fi ≤X

ξ∈Θ

hΥξ(C2C1)f, fi ≤Bhf, fi, ∀f ∈ H,

Hence,{Υξ}_ξ∈Θ is a C2C1-controlled g-frame forHwith respect to{Kξ}_ξ∈Θ Lemma 3.5. Let C1, C2∈GL⁺(H). Then the following statements are equivalent:

(1) {Υξ}_ξ∈Θ is a(C1, C2)-controlled g-frame forH with respect to{Kξ}_ξ∈Θ (2) {vξ,k}_{ξ∈Θ,k∈K}

ξ is a (C1, C2)-controlled forH, where vξ,k= Υ^∗_ξeξ,k, forξ∈Θand k∈ Kξ

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Proof. Let {e_ξ,k}_k∈K

ξ be an orthonormal basis for K_ξ for each ξ ∈ Θ, consequently for any h∈ H, we have Υξh∈ Kξ. It follows that

ΥξC1h= X

k∈Kξ

hΥξC1h, eξ,kieξ,k= X

k∈Kξ

h, C1Υ^∗_ξeξ,k

eξ,k.

and

ΥξC2h= X

k∈Kξ

hΥξC2h, eξ,kieξ,k= X

k∈Kξ

h, C2Υ^∗_ξeξ,k

eξ,k.

We have

hΥξC₁h,Υ_ξC₂hi= X

k∈Kξ

h, C₁Υ^∗_ξe_ξ,k C₂Υ^∗_ξe_ξ,K, h

= X

k∈Kξ

hh, C1v_ξ,ki hC2v_ξ,k, hi.

Hence

Ahh, hi ≤X

ξ∈Θ

hΥξC1h,ΥξC2hi=X

ξ∈Θ

X

k∈Kξ

hh, C1vξ,ki hC2vξ,k, hi ≤Bhh, hi

is equivalent to

Ahh, hi ≤X

ξ∈Θ

X

k∈Kξ

hh, C1vξ,ki hC2vξ,k, hi ≤Bhh, hifor anyh∈ H.

Lemma 3.6. Let C₁, C₂∈GL⁺(H). Then the following assertions are equivalent:

(1) {Υξ}_ξ∈Θ is a(C1, C2)-controlledg-frame forHwith respect to{Kξ}_ξ∈Θ. (2) {C1vξ,k}_{ξ∈Θ,k∈K}

ξ is aC2C₁⁻¹-controlled for H, where vξ,k= Υ^∗_ξeξ,k, forξ∈Θand k∈ K_ξ.

Proof. By the proof of Lemma 3.5, we have X

ξ∈Θ

hΥξC₁f,Υ_ξC₂fi=X

ξ∈Θ

X

k∈kξ

f, C₁Υ^∗_ξe_ξ,k C₂Υ^∗_ξe_ξ,K, f .

Let’s putf_ξ,k=C₁v_ξ,k, v_ξ,k= Υ^∗_ξe_ξ,k, so Ahf, fi ≤X

ξ∈Θ

hΥξC1f,ΥξC2fi ≤Bhf, fi

is equivalent to

Ahf, fi ≤X

ξ∈Θ

X

k∈Kξ

hf, f_ξ,ki

C₂C₁⁻¹v_ξ,k, f

≤Bhf, fifor anyf ∈ H.

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4. Controlled dual g-frames in HilbertC^∗−modules

Definition 4.1. Let C1, C2 ∈ GL⁺(H), {Υξ}_ξ∈Θ and {Φξ}_ξ∈Θ be (C1, C1)-controlled and (C2, C2)-controlledg-Bessel sequences forHwith respect to{Kξ}_ξ∈Θ, respectively. If for every h∈ H,

h=X

ξ∈Θ

C1Υ^∗_ξΦξC2h

Then{Φξ}_ξ∈Θ is called a (C1, C2)-controlled dualg-frame of{Υξ}_ξ∈Θ.

Definition 4.2. Let C1, C2 ∈ GL⁺(H), {Υξ}_ξ∈Θ and {Φξ}_ξ∈Θ be (C1, C1)-controlled and (C2, C2)-controlled g Bessel sequence for H with respect to {Kξ}_ξ∈Θ, respectively. if for any h∈ H,

S_C₁_ΥΦC₂h=X

ξ∈Θ

C₁Υ^∗_ξΦ_ξC₂h.

S_C₁_ΥΦC₂is called a (C₁, C₂)-controlled dual g-frame operator for this pair of controlledg-Bessel sequence.

We clearly see that{Υξ}_ξ∈Θ and {Φξ}_ξ∈Θ are also two g-Bessel sequences. S_C₁_ΥΦC₂ is a well-defined and bounded , and we have

SC₁ΥΦC₂ =TC₁ΥC₁T_C^∗₂_ΦC₂ =C1TΥT_Φ^∗C2=C1SΥΦC2,

Proposition 4.3. Let C1, C2 ∈ GL⁺(H), {Υξ}_ξ∈Θ and {Φξ}_ξ∈Θ be (C1, C1)-controlled and (C2, C2)-controlledg-Bessel sequences with boundsBΥ andBΦ, respectively.

IfS_C₁_ΥΦC₂ is bounded below, then{Υξ}_ξ∈Θ and{Φξ}_ξ∈Θ are(C₁, C₁)-controlled and(C₂, C₂)- controlled g-frames, respectively.

Proof. Let us assume there is a constantλ >0 such that kSC₁ΥΦC₂fk ≥λhf, fifor allf ∈ H.

By Cauchy-Schwartz’s inequality, we have λkhf, fik¹² ≤ kSC1ΥΦC2fk= sup

kgk=1

* X

ξ∈Θ

C1Υ^∗_ξΦξC2f, g +

= sup

kgk=1

X

ξ∈Θ

hΦξC2f,ΥξC1gi

≤ sup

kgk=1

v u u u t

X

ξ∈Θ

hΦξC2f,ΦξC2fi

v u u u t

X

ξ∈Θ

hΥξC1g,ΥξC1gi

≤p BΥ

v u u u t

X

ξ∈Θ

hΦξC2f,ΦξC2fi .

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Thus

λ²

B_Υhf, fi ≤X

ξ∈Θ

hΦ_ξC₂f,Φ_ξC₂fiforf ∈ H On the other hand, Since

S_C^∗₁_ΥΦC₂= (C1SΥΦC2)^∗=C2S^∗_ΥΦC1=C2SΦΥC1=SC₂ΦΥC₁, thenS_C₂_Φ∧C₁ is also bound below. Similarly, we can prove that

λ² BΦ

hf, fi ≤X

ξ∈Θ

hΥξC₂f,Υ_ξC₂fiforf ∈ H

Hence we conclude that{Υ_ξ}_ξ∈Θ is a (C₁, C₁)-controlled g-frames.

Theorem 4.4. Let C₁, C₂ ∈ GL⁺(H), {Υξ}_ξ∈Θ and {Φξ}_ξ∈Θ be (C₁, C₁)-controlled and (C₂, C₂)-controlled g Bessel sequences for H with respect to {K_ξ}_ξ∈Θ, respectively. Then the following statements are equivalent:

(1) f =P

ξ∈ΘC1Υ^∗_ξΦξC2f,∀f ∈ H.

(2) f =P

ξ∈ΘC₂Φ^∗_ξΥ_ξC₁f,∀f ∈ H.

(3) hf, gi=P

ξ∈ΘhΥξC₁f,Φ_ξC₂gi=P

ξ∈ΘhΦξC₂f,Υ_ξC₁gi,∀f, g∈ H.

(4) hf, fi=Pξ∈Θ

ξ∈ΘhΥ_ξC₁f,Φ_ξC₂fi=Pξ∈Θ

ξ∈ΘhΦ_ξC₂f,Υ_ξC₁fi,∀f ∈ H.

In case the equivalent conditions are satisfied, {Υξ}_ξ∈Θ and {Φξ}_ξ∈Θ are (C1, C1)-controlled and(C2, C2)controlled g-frames, respectively.

Proof. (1)⇔(2). LetTC₁ΥC₁ andTC₁ΦC₁ be the synthesis operator of the (C1, C1)-controlled g-Bessel sequence{Υξ}_ξ∈Θ and (C2, C2)-controlled g-Bessel sequence{Φξ}_ξ∈Θrespectively.

Moreover, we see thatTC1AC1T_C^∗₂_ΦC₂=IH, wich is equivalent toTC2ΦC2T_C^∗₁_AC₁=IH, which is identical to the statement (2). Conversely, (2) implies (1) similarly.

(2)⇒(3). suppose that for anyf, g∈ Hwe havef =P

ξ∈ΘC₂Φ^∗_ξΥ_ξC₁f,then hf, gi=hX

ξ∈Θ

C2Φ^∗_ξΥξC1f, gi=X

ξ∈Θ

hΥξC1f,ΦξC2gi=X

ξ∈Θ

hΦξC2f,ΥξC1gi

(2)⇐(3). suppose that for anyf, g∈ H,hf, gi=P

ξ∈ΘhΥξC1f,ΦξC2gishows that

* f−X

ξ∈Θ

C₂Φ^∗_ξΥ_ξC₁f, g +

= 0, ∀g∈ H

Hence (2) is followed.

(3)⇒(4) is evident .

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(4)⇒(3). using condition (4), we have hf+g, f+gi=X

ξ∈Θ

hΥξC1(f+g),ΦξC2(f+g)i

=X

ξ∈Θ

hΥξC1f+ ΥξC1g,ΦξC2f+ ΦξC2gi

=X

ξ∈Θ

hΥ_ξC₁f,Φ_ξC₂fi+X

ξ∈Θ

hΥ_ξC₁f,Φ_ξC₂gi

+X

ξ∈Θ

hΥξC1g,ΦξC2fi+X

ξ∈Θ

hΥξC1g,ΦξC2gi. Also,

hf −g, f−gi=X

ξ∈Θ

hΥξC1f,ΓξC2fi −X

ξ∈Θ

hΥξC1f,ΦξC2gi

−X

ξ∈Θ

hΥ_ξC₁g,Φ_ξC₂fi+X

ξ∈Θ

hΥ_ξC₁g,Φ_ξC₂gi. hf + ig, f+ igi=X

ξ∈Θ

hΥξC1f,ΦξC2fi −iX

ξ∈Θ

hΥξC1f,ΦξC2gi + iX

ξ∈Θ

hΥ_ξC₁g,Φ_ξC₂gi. hf−ig, f−igi=X

ξ∈Θ

hΥξC1f,ΦξC2fi+ iX

ξ∈Θ

hΥξC1f,ΦξC2gi

−iX

ξ∈Θ

hΥ_ξC₁g,Φ_ξC₂gi. and from the polarization identity,

hf, gi=1

4(hf+g, f+gi − hf−g, f−gi+ ihf + ig, f+ igi −ihf−ig, f−igi)

=X

ξ∈Θ

hΥξC1f,ΦξC2gi.

Lemma 4.5. [15] LetC1, C2∈GL⁺(H), the operator TC1ΥC2 :M

ξ∈Θ

Kξ → H, TC1ΥC2

{fξ}_ξ∈Θ

=X

ξ∈Θ

pC₁C₂Υ^∗_ξf_ξ

is well-defined and bounded with kTC₁ΥC₂k ≤ √

B. If and only if {Υξ:ξ∈Θ} is (C1, C2)- controlled g-Bessel sequence forHwith respect to{Kξ}_ξ∈Θ with bound B.

Theorem 4.6. LetC₁, C₂∈GL⁺(H). A sequence{Υξ:ξ∈Θ} ⊂End^∗_A(H,Kξ)be a(C₁, C₁)- controlled g-frame forHwith respect to{Kξ}_ξ∈Θ. Then the following statements are equivalent:

(1) a(C2, C2)-controlledg-frame{Φξ}_ξ∈Θis a(C1, C2)-controlled dualg-frame of{Υξ}_ξ∈Θ (2) C2Φ^∗_ξeξ,k =U(eξ,kδξ), ξ ∈Θ, k ∈ Kξ ⊂Z, where U : L

ξ∈ΘKξ → H is a bounded left-inverse ofT_C^∗

1∧C1.

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Proof. We suppose that{k_ξ}_ξ∈Θ∈L

ξ∈ΘK_ξ, thus

(4.1) {kξ}_ξ∈Θ=X

ξ∈Θ

kξδξ=X

ξ∈Θ

X

k∈K_ξ

hkξ, eξ,kieξ,kδξ. whereδis the Kronecker symbol.

We have{eξ,kδξ}_{ξ∈Θ,k∈K}

ξis an orthonormal basis ofL

ξ∈ΘKξ. If there existU :L

ξ∈ΘKξ→ H is a bounded left-inverse ofT_C^∗

1ΥC₁ such that

U(eξ,kδξ) =C2Φ^∗_ξeξ,k, ξ∈Θ, k∈Kξ.

applying Lemma 4.1,{Φξ}_ξ∈Θ is a (C₂, C₂)-controlled g-Bessel sequence for Hwith respect to {K_ξ}_ξ∈Θ. For every h∈ H, the equation (4,1) gives us

h=U T_C^∗

1ΥC₁h=U



 X

ξ∈Θ

X

k∈Kξ

hΥ_ξC₁h, e_ξ,kie_ξ,kδ_ξ





=X

ξ∈Θ

X

k∈Kξ

h, C₁Υ^∗_ξe_ξ,k

U(e_ξ,kδ_ξ)

=X

ξ∈Θ

C2Φ^∗_ξ X

k∈Kξ

hh, C1vξ,kieξ,k

=X

ξ∈Θ

C2Φ^∗_ξΥξC1h,

wherevξ,k= Υ^∗_ξeξ,k. we have,{Φξ}_ξ∈Θis a (C1, C2)-controlled dual g-frame of{Υξ}_ξ∈Θ Conversely, Forh∈ H, we have

h=X

ξ∈Θ

C1Υ^∗_ξΦξC2h=X

ξ∈Θ

C2Φ^∗_ξΥξC1h which isTC₂ΦC₂T_C^∗₁_ΥC₁ =IH. LetU =TC₂ΦC₂, thenU :L

ξ∈ΘKξ→ His a bounded left-inverse ofT_C^∗₁_ΥC₁. Then

h=X

ξ∈Θ

X

k∈Kξ

hh, C₁v_ξ,kiC₂Φ^∗_ξe_ξ,k=X

ξ∈Θ

X

k∈Kξ

hh, C₁v_ξ,ki U(e_ξ,kδ_ξ), ∀h∈ H since{e_ξ,k}_k∈K

ξ is an orthonormal basis ofK_ξ, we have

C2Φ^∗_ξeξ,k=U(eξ,kδξ), ξ∈Θ, k∈Kξ.

Theorem 4.7. Let∆∈GL⁺(H),{Υξ}_ξ∈Θbe a(∆,∆)-controlled g-frame forHwith respect to {K_ξ}_ξ∈Θ with the frame operator and synthesis operator S_∆Υ∆ and T_∆Υ∆, respectively. Then A sequence{Φ_ξ:ξ∈Θ} ⊂End^∗_A(H,K_ξ)is a∆-controlled dualg-frame of{Υ_ξ}_ξ∈Θif and only if

Φ_ξh= (Th)_ξ+ Υ_ξS_∆Υ∆⁻¹ ∆h, ξ∈Θ, h∈ H whereT :H →L

ξ∈ΘKξ is a bounded linear operator satisfyingT∆Υ∆T = 0

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Proof. If T : H → L

ξ∈ΘK_ξ is a bounded linear operator satisfying T_∆Υ∆T = 0. Then {Φξ :ξ∈Θ} ⊂End^∗_A(H,Kξ) is a g-Bessel sequence forHwith respect to{Kξ}_ξ∈Θ. In fact, for anyh∈ Hwe have

X

ξ∈Θ

hΦξh,Φξhi=X

ξ∈Θ

h(Th)ξ+ ΥξS_∆Υ∆⁻¹ ∆h,(Th)ξ+ ΥξS_∆Υ∆⁻¹ ∆hi

≤2



 X

ξ∈Θ

hΥ_ξS_∆Υ∆⁻¹ P f,Υ_ξS_∆Υ∆⁻¹ ∆hi+kThk²





≤2



 X

ξ∈Θ

hΥ_ξS_∆Υ∆⁻¹ ∆,Υ_ξS_∆Υ∆⁻¹ ∆ihh, hi+kT k²hh, hi





≤2 B

S_∆Υ∆⁻¹ ∆

2+kT k² hh, hi, whereB is the upper bound of{Υξ}_ξ∈Θ. Furthermore,

X

ξ∈Θ

∆Υ^∗_ξΦ_ξh=X

ξ∈Θ

∆Υ^∗_ξ (Th)_ξ+ Υ_ξS_∆Υ∆⁻¹ ∆h

=T∆Υ∆Th+X

ξ∈Θ

∆Υ^∗_ξΥ_ξS_∆Υ∆⁻¹ ∆h

= 0 +S⁻¹_∆Υ∆X

ξ∈Θ

∆Υ^∗_ξΥξ∆h

=h.

Thus{Φξ:ξ∈Θ} ⊂ End^∗_A(H,Kξ) is a ∆-controlled dual g-frame of {Υξ}_ξ∈Θ. On the other hand. Suppose that {Φξ ∈End^∗_A(H,Kξ) :ξ∈Θ} is a ∆-controlled dual g-frame of{Υξ}_ξ∈Θ. Now we consider the operatorT which is defined by

T :H →M

ξ∈Θ

Kξ, h7→Sh (∀h∈ H)

satisfying

Φ_ξh= (Th)_ξ+ Υ_ξS_∆Υ∆⁻¹ ∆h, ξ∈Θ.

Hence

kThk²=X

ξ∈Θ

hΦξh−ΥξS_∆Υ∆⁻¹ ∆h,Φξh−ΥξS_∆Υ∆⁻¹ ∆hi

≤X

ξ∈Θ

hΦξh,Φξhi+X

ξ∈Θ

hΥξS_∆Υ∆⁻¹ ∆h,ΥξS_∆Υ∆⁻¹ ∆hi

+ 2



 X

ξ∈Θ

hΦξh,Φξhi





1 2

 X

ξ∈Θ

hΥξS_∆Υ∆⁻¹ ∆h,ΥξS_∆Υ∆⁻¹ ∆hi





1 2

≤

C+D⁻¹+ 2√ CD⁻¹

hh, hi,

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thenT is a linear bounded operator. Furthermore, for anyh, g∈ H, we have hT_∆Υ∆Th, gi=X

ξ∈Θ

∆Υ^∗_ξTh, g

=X

ξ∈Θ

∆Υ^∗_ξ Φ_ξh−Υ_ξS_∆Υ∆⁻¹ ∆h , g

=X

ξ∈Θ

∆Υ^∗_ξΦξh, g

−X

ξ∈Θ

∆Υ^∗_ξΥξS_∆Υ∆⁻¹ ∆h, g

=hh, gi − hh, gi= 0.

Hence we conclude thatT∆Υ∆T = 0.

Declarations

Availablity of data and materials Not applicable.

Competing interest

The authors declare that they have no competing interests.

Fundings

Authors declare that there is no funding available for this article.

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.

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