View of Tensor product for g-fusion frame in hilbert $C^{\ast}-$modules

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Tensor product for g-fusion frame in Hilbert C

^∗

−modules

Fakhr-dine Nhari¹, Mohamed Rossafi², Youssef Aribou³

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

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

3Laayoune Higher School of Technology, Ibn Zohr University, Laayoune, Morocco

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

Abstract.

In this paper, we stady the tensor product ofg−fusion frame in HilbertC^∗−modules and we give the frame operator for a pair ofg−fusion Bessel sequences in tensor product of Hilbert C^∗−modules.

Keywords: G-fusion Frame, tensor product, Hilbert C^∗−module.

1. Introduction and Preliminaries

In the study of vector spaces one of the most important concepts is that of a basis, allowing each element in the space to be written as a linear combination of the elements in the basis.

However, the conditions to a basis are very restrictive: linear independence between the elements. This makes it hard or even impossible to find bases satisfying extra conditions, and this is the reason that one might look for a more flexible substitute. Frames are such tools.

A frame for a vector space equipped with inner product also allows each element in the space to be written as a linear combination of the elements in the frame, but linear independence between the frame elements is not required.

Frames for Hilbert spaces were introduced by Duffin and schaefer [5] in 1952 to study some deep problems in nonharmonic fourier series by abstracting the fondamental notion of Gabor [6] for signal processing.

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Tensor product for g-fusion frame in HilbertC^∗−modules 137

Many generalizations of the concept of frame have been defined in HilbertC^∗-modules [7,9, 11,12,13,14,15]

Tensor products have important applications, for example tensor products are useful in the approximation of multi-variate functions of combinations of univariate ones.

In this paper we consider the tensor product of HilbertC^∗−modules and we generalize some of known results about frames to generalized-fusion frames in HilbertC^∗−modules.

The paper is organized as follows, we continue this introductory section we briefly recall the definitions and basic properties of C^∗−algebra and Hilbert C^∗−modules. In section 2, we introduce the concept ofg−fusion frame, and gives an equivalent definition. In section 3, we introduce the concept of the tensor product of g−fusion frame and gives some properties.

Finally in section 4, we discuss the frame operator for a pair ofg−fusion Bessel sequences in tensor product of HilbertC^∗−modules.

In the following we briefly recall the definitions and basic properties ofC^∗-algebra, Hilbert A-modules. Our reference forC^∗-algebras is [4,3]. For aC^∗-algebraAifa∈ A is positive we writea≥0 andA⁺ denotes the set of positive elements ofA.

Definition 1.1. [3]. IfAis a Banach algebra, an involution is a mapa→a^∗ ofAinto itself such that for allaandb inAand all scalarsαthe following conditions hold:

(1) (a^∗)^∗=a.

(2) (ab)^∗=b^∗a^∗.

(3) (αa+b)^∗= ¯αa^∗+b^∗.

Definition 1.2. [3]. AC^∗-algebraAis a Banach algebra with involution such that : ka^∗ak=kak²

for everyainA.

Example 1.3. B=B(H) the algebra of bounded operators on a Hilbert space, is aC^∗-algebra, where for each operatorA,A^∗ is the adjoint of A.

Definition 1.4. [8]. LetA be a unital C^∗-algebra andH be a left A-module, such that the linear structures ofA and U are compatible. H is a pre-Hilbert A-module if H is equipped with anA-valued inner producth., .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∈H andhx, 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.

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138 Fakhr-dine Nhari, Mohamed Rossafi and Youssef Aribou

For x ∈ H, we define ||x|| = ||hx, xi||¹². If H is complete with ||.||, it is called a Hilbert A-module or a HilbertC^∗-module overA.

Throughout this paper I and J are finite or countably infinite index sets, we fix the nota- tionsAand B for a given unitalC^∗−algebras, H and K are the countably generated Hilbert A−module andB−module, respectively. Let{Hi}_i∈I and{Kj}_j∈J are the sequences of closed orthogonally complemented submodules of H and K, respectively. {Wi}i∈I and {Vj}j∈J are the sequences of HilbertC^∗−modules. End^∗_A(H, Wi) is a set of all adjointable operator from H toW_i. In particularEnd^∗_A(H) denote the set of all adjointable operators onH. P_H_i denote the orthogonal projection onto the closed submodule orthogonally complementedH_i ofH.

Define the HilbertA−module

l²({Wi}i∈I) ={{xi}i∈I :xi ∈Wi,kX

i∈I

hxi, xiik<∞}

withA−valued inner producth{xi}_i∈I,{yi}_i∈Ii=P

i∈Ihxi, yii, where{xi}_i∈I,{y}_i∈I ∈l²({Wi}_i∈I) Lemma 1.5. [2]. Let H and K 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 ism >0such thatkT^∗xk ≥mkxk for allx∈K.

(iii) T^∗ is bounded below with respect to the inner product, i.e., there is m⁰ >0 such that hT^∗x, T^∗xi ≥m⁰hx, xifor all x∈K.

Lemma 1.6. [1]. LetU andH two HilbertA-modules andT ∈End^∗_A(U, H). Then:

(i) IfT is injective andT has closed range, then the adjointable mapT^∗T is invertible and k(T^∗T)⁻¹k⁻¹≤T^∗T ≤ kTk².

(ii) IfT is surjective, then the adjointable mapT T^∗ is invertible and k(T T^∗)⁻¹k⁻¹≤T T^∗≤ kTk².

Lemma 1.7. [2] Let H be a Hilbert A-module over aC^∗-algebraA, and T ∈End^∗_A(H) such thatT^∗=T. The following statements are equivalent:

(i) T is surjective.

(ii) There arem, M >0 such thatmkxk ≤ kT xk ≤Mkxk, for allx∈H.

(iii) There arem⁰, M⁰ >0 such that m⁰hx, xi ≤ hT x, T xi ≤M⁰hx, xifor allx∈H.

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2. g−fusion frame in HilbertC^∗−modules We begin this section with the following lemma:

Lemma 2.1. Let {Hi}_i∈I be a sequence of orthogonally complemented closed submodules of H andT ∈ End^∗_A(H) invertible, if T^∗T Hi ⊆Hi for each i∈I, then{T Hi}_i∈I is a sequence of orthogonally complemented closed submodules andPH_iT^∗=PH_iT^∗PT H_i.

Proof. Firstly for each i∈I, T :Hi →T Hi is invertible, so each T Hi is a closed submodule of H. We show that H =T Hi⊕T(H_i^⊥). SinceH =T H, then for each x∈H, there exists y ∈ H sutch that x = T y. On the other hand y = u+v, for some u ∈ H_i and v ∈ H_i^⊥. Hencex=T u+T v, whereT u∈T H_iandT v∈T(H_i^⊥), plainlyT H_i∩T(H_i^⊥) = (0), therefore H = T H_i ⊕T(H_i^⊥). Hence for every y ∈ H_i, z ∈ H_i^⊥ we have T^∗T y ∈ H_i and therefore hT y, T zi = hT^∗T y, zi = 0, so T(H_i^⊥) ⊂ (T Hi)^⊥ and consequently T(H_i^⊥) = (T Hi)^⊥ witch implies thatT Hi is orthogonally complemented.

Letx∈Hwe havex=PT H_ix+y, for somey∈(T Hi)^⊥, thenT^∗x=T^∗PT H_ix+T^∗y. Letv∈Hi

thenhT^∗y, vi=hy, T vi= 0 thenT^∗y ∈H_i^⊥ and we havePH_iT^∗x=PH_iT^∗PT H_ix+PH_iT^∗y, thenPH_iT^∗x=PH_iT^∗PT H_ixthus implies that for eachi∈Iwe havePH_iT^∗=PH_iT^∗PT H_i.

Definition 2.2. Let{Hi}i∈I be a sequence of closed orthogonally complemented submodules ofH, {vi}i∈I be a familly of positive weights inA, i.e., eachv_i is a positive invertible element from the center of the C^∗−algebra A and Λ_i ∈ End^∗_A(H, W_i) for all i ∈ I. We say that Λ ={H_i,Λ_i, v_i}_i∈I is ag−fusion frame forH if and only if there exists two constants 0< A≤ B <∞such that

(2.1) Ahx, xi ≤X

i∈I

v_i²hΛ_iP_H_ix,Λ_iP_H_ixi ≤Bhx, xi, ∀x∈H.

The constantsAandBare called the lower and upper bounds ofg−fusion frame, respactively.

IfA =B then Λ is called tight g-fusion frame and ifA =B = 1 then we say Λ is a Parseval g−fusion frame. If Λ satisfies the inequality

X

i∈I

v_i²hΛiPH_ix,ΛiPH_ixi ≤Bhx, xi, ∀x∈H.

then it is called ag−fusion Bessel sequence with boundB in H.

Lemma 2.3. letΛ = (Hi,Λi, vi)_i∈I be ag−fusion Bessel sequence forH with boundB. Then for each sequence{xi}_i∈I ∈l²({Wi}_i∈I), the seriesP

i∈IviPH_iΛ^∗_ixiis converge unconditionally.

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Proof. letJ be a finite subset ofI, then

||X

i∈J

viPHiΛ^∗_ixi||= sup

||y||=1

||hX

i∈J

viPHiΛ^∗_ixi, yi||

≤ ||X

i∈J

hxi, x_ii||¹² sup

||y||=1

||X

i∈J

v_i²hΛiP_H_iy,Λ_iP_H_iyi||¹²

≤√ B||X

i∈J

hxi, x_ii||¹². And it follows thatP

j∈IvjPW_jΛ^∗_jfj is unconditionally convergent inH. Now, we can define the synthesis operator by lemma2.3

Definition 2.4. let Λ = (Hi,Λi, vi)_i∈I be ag−fusion Bessel sequence forH. Then the operator TΛ:l²({Wi}_i∈I)→H defined by

T_Λ({x_i}_i∈I) =X

i∈I

v_iP_W_iΛ^∗_ix_i, ∀{x_i}_i∈I ∈l²({W_i}_i∈I).

Is called synthesis operator. We say the adjoint T_Λ^∗ of the synthesis operator the analysis operator and it is defined byT_Λ^∗:H →l²({Wi}i∈I) such that

T_Λ^∗(x) ={viΛiPH_i(x)}_i∈I, ∀x∈H.

The operatorS_Λ:H →H defined by SΛx=TΛT_Λ^∗x=X

j∈I

v²_iPH_iΛ^∗_iΛiPH_i(x), ∀x∈H.

Is calledg−fusion frame operator. It can be easily verify that

(2.2) hSΛx, xi=X

i∈I

v_i²hΛiPH_i(x),ΛiPH_i(x)i, ∀x∈H.

Furthermore, if Λ is ag−fusion frame with boundsAandB, then Ahx, xi ≤ hSΛx, xi ≤Bhx, xi, ∀x∈H.

It easy to see that the operator SΛ is bounded, self-adjoint, positive, now we proof the inversibility ofSΛ. Letx∈H we have

||T_Λ^∗(x)||=||{v_iΛ_iP_W_i(x)}_i∈I||=||X

i∈I

v²_ihΛ_iP_H_i(x),Λ_iP_H_i(x)i||¹². Since Λ isg−fusion frame then

√A||hx, xi||¹² ≤ ||T_Λ^∗x||.

Then

√

A||x|| ≤ ||T_Λ^∗x||.

Frome lemma 1.5, TΛ is surjective and by lemma 1.6, TΛT_Λ^∗ = SΛ is invertible. We now, AIH≤SΛ≤BIH and this givesB⁻¹IH ≤S_Λ⁻¹≤A⁻¹IH

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Theorem 2.5. Let H be a Hilbert A−module overC^∗−algebra. ThenΛ = (H_i,Λ_i, v_i)_i∈I is a g−fusion frame forH if and only if there exist two constants0< A≤B <∞such that for all x∈H

A||hx, xi|| ≤ ||X

i∈I

v²_ihΛiPH_ix,ΛiPH_ixi|| ≤B||hx, xi||.

Proof. Suppose Λ isg−fusion frame forH, then for allx∈H, A||hx, xi|| ≤ ||X

i∈I

v_i²hΛiPH_ix,ΛiPH_ixi|| ≤B||hx, xi||

Conversely, for eachx∈H we have

||X

i∈I

v_i²hΛiPH_ix,ΛiPH_ixi||=||X

i∈I

hviΛiPH_ix, viΛiPH_ixi||

=||h{viΛiPH_ix}_i∈I,{viΛiPH_ix}_i∈Ii||

=||{viΛiPH_ix}i∈I||².

We define the operatorL:H → ⊕_i∈IWi byL(x) ={viΛiPH_ix}_i∈I, then

||L(x)||²=||(viΛiPH_ix)i∈I||²≤B||x||². LisA−linear bounded operator, then there existC >0 sutch that

hL(x), L(x)i ≤Chx, xi, ∀x∈ H.

So

X

j∈I

v_i²hΛiPH_ix,ΛiPH_ixi ≤Chx, xi, ∀x∈H.

Therefore Λ is a g−fusion Bessel sequence for H. Now we cant define the g−fusion frame operatorSΛ onH. So

X

j∈J

v²_jhΛjPWjx,ΛjPWjxi=hSΛx, xi, ∀x∈H.

SinceS_Λ is positive, self-adjoint, then hS_Λ¹²x, S

1 2

Λxi=hSΛx, xi, ∀x∈H.

That implies

A||hx, xi|| ≤ ||hS_Λ¹²x, S_Λ¹²xi|| ≤B||hx, xi||, ∀x∈H.

Frome lemma1.7there exist two canstantsA⁰, B⁰ >0 such that A⁰hx, xi ≤ hS_Λ¹²x, S

1 2

Λxi ≤B⁰hx, xi, ∀f ∈H.

So

A⁰hx, xi ≤X

i∈I

v_i²hΛiPH_ix,ΛiPH_ixi ≤B⁰hx, xi, ∀x∈H.

Hence Λ is ag−fusion frame forH.

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3. Tensor product ofg−fusion frame in HilbertC^∗−modules

Suppose thatA,Bare unitalsC^∗−algebras and we takeA ⊗ Bas the completion ofA ⊗algB with the spatial norm. A ⊗ B is the spatial tensor product of Aand B, also suppose that H is a HilbertA−module andK is a HilbertB−module. We want to defineH⊗K as a Hilbert (A ⊗ B)−module. Start by forming the algebraic tensor productH⊗algK of the vector spaces H, K (over C). This is a left module overh (A ⊗alg B) (the module action being given by (a⊗b)(x⊗y) =ax⊗by(a∈ A, b∈ B, x∈H, y∈K)). For (x1, x2 ∈H, y1, y2∈K) we define hx1⊗y1, x₂⊗y2iA⊗B=hx1, x₂iA⊗hy1, y₂iB. We also know that forz=Pn

i=1x_i⊗yiinH⊗algK we havehz, ziA⊗B=P

i,jhxi, x_jiA⊗ hyi, y_jiB≥0 and hz, ziA⊗B= 0 iffz= 0. This extends by linearity to an (A ⊗_algB)−valued sesquilinear from onH⊗_algK, which makesH⊗_algK into a semi-inner-product module over the pre-C^∗−algebra (A ⊗_algB). The semi-inner-product on H⊗algKis actually an inner product. see [10]. ThenH⊗algKis an inner-product module over the pre-C^∗−algebra (A ⊗algB), and we can perform the double completion discussed in chapter 1 of [10] to conclude that the completionH⊗K ofA ⊗algBis a Hilbert (A ⊗ B)−module. We callH⊗Kthe exterior tensor product ofH andK. WithH, Kas above, we wish to investigate the adjointable operators onH⊗K. Suppose thatS ∈End^∗_A(H) andT ∈End^∗_B(K). We define a linear operatorS⊗T onH⊗K byS⊗T(x⊗y) =Sx⊗T y (x∈H, y∈K). It is a routine verification that is S^∗⊗T^∗ is the adjoint ofS⊗T, so in factS⊗T ∈End^∗_A⊗B(H⊗K). We note that ifa∈ A⁺ andb∈ B⁺, thena⊗b∈(A ⊗ B)⁺. Plainly ifa, bare Hermitian elements ofAanda≤b, then for every positive elementxofB, we havea⊗x≥b⊗x.

Definition 3.1. Let{vi}i∈I,{wj}j∈J be two families of positive weights, i.e., each v_i andw_j are positive invertible elements of A, and Λ_i⊗Γ_j ∈ End^∗_A(H ⊗K, W_i⊗V_j) for each i ∈ I and j ∈J. Then the family Λ⊗Γ = {H_i⊗K_j,Λ_i⊗Γ_j, v_iw_j}_i,j is saide to be a generalized fusion frame org−fusion frame forH⊗Kwith respect to{Hi⊗Kj}i,j if there exist constants 0< A≤B <∞such that for allx⊗y∈H⊗K

Ahx⊗y, x⊗yi ≤X

i,j

v_i²w_j²h(Λi⊗Γj)PH_i⊗Kj(x⊗y),(Λi⊗Γj)PH_i⊗Kj(x⊗y)i ≤Bhx⊗y, x⊗yi

where PH_i⊗Kj is the orthogonal projection of H⊗K ontoHi⊗Kj. The constants A andB are called the frame bounds of Λ⊗Γ. IfA=B then it is called a tightg−fusion frame. If the family Λ⊗Γ satisfies the inequality, for eachx⊗y∈H⊗K

X

i,j

v_i²w_j²h(Λi⊗Γj)PH_i⊗Kj(x⊗y),(Λi⊗Γj)PH_i⊗Kj(x⊗y)i ≤Bhx⊗y, x⊗yi

then it is called ag−fusion Bessel sequence inH⊗Kwith boundB.

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Definition 3.2. Fori∈I andj∈J, define the HilbertA ⊗ B−module l²({W_i⊗V_j}) ={{x_i⊗y_j}:x_i⊗y_j∈W_i⊗V_j,kX

i,j

hx_i⊗y_j, x_i⊗y_jik<∞}

with theA ⊗ B−valued inner product h{xi⊗y_j},{x⁰_i⊗y_j⁰}i=X

i,j

hxi⊗y_j, x⁰_i⊗y_j⁰i

=X

i,j

hxi, x⁰_iiWi⊗ hyj, y⁰_jiVj

=

X

i∈I

hxi, x⁰_iiWi

⊗

X

j∈J

hyj, y_j⁰iVj

=h{xi}i∈I,{x⁰_i}i∈Iil²({Wi}i∈I)⊗ h{yj}j∈J,{yj}j∈Jil²({Vj}j∈J). Theorem 3.3. The familiesΛ ={Hi,Λ_i, v_i}i∈I andΓ ={Kj,Γ_j, w_j}j∈J areg−fusion frames forH andK with respect to{W_i}_i∈I and{V_j}_j∈J, respectively if and only if the familyΛ⊗Γ = {H_i⊗K_j,Λ_i⊗Γ_j, v_iw_j}_i,j is ag−fusion frame for H⊗K with respect{W_i⊗V_j}_i,j.

Proof. Suppose that Λ and Γ are g−fusion frame for H and K. Then there exist positive constants (A, B) and (C, D) such that

Akxk²≤ kX

i∈I

v²_ihΛiPH_ix,ΛiPH_ixi_Ak ≤Bkxk²,∀x∈H,

Ckyk²≤ kX

j∈J

w²_jhΓjPK_jy,ΓjPK_jyi_Bk ≤Dkyk²,∀y∈K, then,

ACkxk²kyk²≤ kX

i∈I

v²_ihΛiPH_ix,ΛiPH_ixiAkkX

j∈J

w_j²hΓjPK_jy,ΓjPK_jyiBk ≤BDkxk²kyk²,

hence,

ACkx⊗yk²≤ kX

i∈I

v_i²hΛiPH_ix,ΛiPH_ixi_A⊗X

j∈J

w²_jhΓjPK_jy,ΓjPK_jyi_Bk ≤BDkx⊗yk²,

so,

ACkx⊗yk²≤ kX

i,j

v²_iw²_jhΛiPHix⊗ΓjPKjy,ΛiPHix⊗ΓjPKjyik ≤BDkx⊗yk².

Therefore, for eachx⊗y∈H⊗K ACkx⊗yk²≤ kX

i,j

v_i²w_j²h(Λi⊗Γj)PH_i⊗Kj(x⊗y),(Λi⊗Γj)PH_i⊗Kj(x⊗y)ik ≤BDkx⊗yk², 142143

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we conclude that, Λ⊗Γ is a g−fusion frame forH ⊗K. Conversely, suppose that Λ⊗Γ is a g−fusion frame for H ⊗K, then there exist constants A, B > 0 such that for all x⊗y ∈ H⊗K− {0⊗0},

Akx⊗yk²≤ kX

i,j

v²_iw²_jh(Λi⊗Γ_j)P_H_i_⊗K_j(x⊗y),(Λ_i⊗Γ_j)P_H_i_⊗K_j(x⊗y)ik ≤Bkx⊗yk²,

then,

Akx⊗yk²≤ kX

i∈I

v²_ihΛiPH_ix,ΛiPH_ixiA⊗X

j∈J

w²_jhΓjPK_jy,ΓjPK_jyiBk ≤Bkx⊗yk²,

hence,

Akxk²kyk²≤ kX

i∈I

v_i²hΛiPH_ix,ΛiPH_ixi_AkkX

j∈J

w²_jhΓjPK_jy,ΓjPK_jyi_Bk ≤Bkxk²kyk²,

So,

Akyk² kP

j∈Jw²_jhΓjPK_jy,ΓjPK_jyi_Bkkxk²≤ kX

i∈I

v²_ihΛiPH_ix,ΛiPH_ixiAk,

and

kX

i∈I

v²_ihΛiPH_ix,ΛiPH_ixiAk ≤ Bkyk² kP

j∈Jw²_jhΓjPK_jy,ΓjPK_jyi_Bkkxk².

Therefore, Λ is ag−fusion frame forH. Similarly, it can be schown that Γ is ag−fusion frame

forK.

Definition 3.4. Let Λ⊗Γ ={Hi⊗K_j,Λ_i⊗Γ_j, v_iw_j}i,j be ag−fusion frame forH⊗K. The synthesis operatorT_ΛΓ:l²({Wi⊗V_j})→H⊗K is given by

TΛΓ({xi⊗yj}) =X

i,j

viwjPH_i⊗Kj(Λi⊗Γj)^∗(xi⊗yj), ∀{xi⊗yj} ∈l²({Wi⊗Vj}).

And the frame operatorS_Λ⊗Γ :H⊗K→H⊗Kis described by

S_Λ⊗Γ(x⊗y) =X

i,j

(viwj)²PH_i⊗Kj(Λi⊗Γj)^∗(Λi⊗Γj)PH_i⊗Kj(x⊗y), ∀x⊗y∈H⊗K.

Theorem 3.5. If SΛ, SΓ and S_Λ⊗Γ are the associated g−fusion frame operators and TΛ, TΓ

andT_Λ⊗Γ are the synthesis operators ofg−fusion framesΛ,ΓandΛ⊗ΓforH,KandH⊗K, respectively, thenS_Λ⊗Γ=SΛ⊗SΓ,S_Λ⊗Γ⁻¹ =S⁻¹_Λ ⊗S_Γ⁻¹, and ,T_Λ⊗Γ=TΛ⊗TΓ,T_Λ⊗Γ^∗ =T_Λ^∗⊗T_Γ^∗. 143144

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Proof. Let eachx⊗y∈H⊗K, we have

S_Λ⊗Γ(x⊗y) =X

i,j

v_i²w_j²PH_i⊗Kj(Λi⊗Γj)^∗(Λi⊗Γj)PH_i⊗Kj(x⊗y)

=X

i,j

v_i²w_j²(PH_i⊗Kj)(Λi⊗Γj)^∗(Λi⊗Γj)(PH_i⊗PK_j)(x⊗y)

=X

i,j

v_i²w_j²(PH_iΛ^∗_iΛiPH_ix)⊗(PK_jΓ^∗_jΓjPK_jy)

= X

i∈I

v²_iPH_iΛ^∗_iΛiPH_ix

⊗ X

j∈J

w_j²PK_jΓ^∗_jΓjPK_jy

=SΛx⊗SΓy

= (SΛ⊗SΓ)(x⊗y).

Then,S_Λ⊗Γ=S_Λ⊗S_Γ, so S_Λ⊗Γ⁻¹ =S_Λ⁻¹⊗S_Γ⁻¹.

On the other hand, for each{x_i⊗y_j} ∈l²({W_i⊗V_j}), we have

T_Λ⊗Γ({xi⊗yj}) =X

i,j

viwjPH_i⊗Kj(Λi⊗Γj)^∗(xi⊗yj)

=

X

i∈I

viPH_iΛ^∗_ixi

⊗

X

j∈J

wjPK_jΓ^∗_jyj

=T_Λ({x_i})⊗T_Γ({y_j})

= (TΛ⊗TΓ)({xi⊗yj}).

this shows thatTΛ⊗Γ =TΛ⊗TΓ, henceT_Λ⊗Γ^∗ =T_Λ^∗⊗T_Γ^∗.

Theorem 3.6. LetΛ ={Hi,Λi, vi}_i∈I andΓ ={Kj,Γj, wj}_j∈J beg−fusion frames forH and K with g−fusion frame operators SΛ and SΓ, respectively. Then θ ={S_Λ⊗Γ⁻¹ (Hi⊗Kj),(Λi⊗ Γj)PH_i⊗KjS_Λ⊗Γ⁻¹ , viwj}i,j is ag−fusion frame for H⊗K.

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Proof. Let (A, B) and (C, D) be theg−fusion frame bounds of Λ and Γ, respectively. Now, we have for eachx⊗y∈H⊗K,

kX

i,j

v_i²w²_jh(Λi⊗Γj)PH_i⊗KjS_Λ⊗Γ⁻¹ P_S⁻¹

Λ⊗Γ(Hi⊗Kj)(x⊗y),(Λi⊗Γj)PH_i⊗KjS⁻¹_Λ⊗ΓP_S⁻¹

Λ⊗Γ(Hi⊗Kj)(x⊗y)ik

=kX

i,j

v_i²w²_jhΛi⊗ΓjPH_i⊗KjS_Λ⁻¹⊗S_Γ⁻¹P_S⁻¹

Λ Hi⊗P_S⁻¹

Γ Kj(x⊗y), Λi⊗ΓjPH_i⊗KjS_Λ⁻¹⊗S_Γ⁻¹P_S⁻¹

Λ Hi⊗P_S⁻¹

Γ Kj(x⊗y)ik

=kX

i,j

v_i²w²_jhΛi⊗Γ_jP_H_i⊗P_K_jS_Λ⁻¹P_S−1

Λ H_i⊗S_Γ⁻¹P_S−1

Γ K_j(x⊗y), Λ_i⊗Γ_jP_H_i⊗P_K_jS_Λ⁻¹P_S−1

Λ H_i⊗S_Γ⁻¹P_S−1

Γ K_j(x⊗y)ik

=kX

i,j

v_i²w²_jhΛiPH_iS_Λ⁻¹P_S−1

Λ Hix⊗ΓjPK_jS_Γ⁻¹P_S−1

Γ Kjy,ΛiPH_iS_Λ⁻¹P_S−1

Λ Hix⊗ΓjPK_jS_Γ⁻¹P_S−1 Γ Kjyik

=kX

i,j

v_i²w²_jhΛiPH_iS_Λ⁻¹x⊗ΓjPK_jS_Γ⁻¹y,ΛiPH_iS⁻¹_Λ x⊗ΓjPK_jS_Γ⁻¹yik

=kX

i,j

v_i²w²_jhΛiPH_iS_Λ⁻¹x,ΛiPH_iS_Λ⁻¹xi_A⊗ hΓjPK_jS_Γ⁻¹y,ΓjPK_jS_Γ⁻¹yi_Bk

=kX

i∈I

v_i²hΛiPH_iS_Λ⁻¹x,ΛiPH_iS_Λ⁻¹xi_AkkX

i∈J

w²_jhΓjPK_jS_Γ⁻¹y,ΓjPK_jS⁻¹_Γ yi_Bk

≤BkS_Λ⁻¹xk²DkS_Γ⁻¹yk²

≤ BD

(AC)²kx⊗yk².

On the other hand for eachx⊗y∈H⊗K,

kx⊗yk⁴=khx, xiAk²khy, yiBk²

=khX

i∈I

v_i²P_H_iΛ^∗_iΛ_iP_H_iS_Λ⁻¹x, xiAk²khX

i∈J

w_j²P_K_jΓ^∗_jΓ_jP_K_jS_Γ⁻¹y, yiBk²

=kX

i∈I

v_i²hΛ_iP_H_iS⁻¹_Λ x,Λ_iP_H_ixi_Ak²kX

j∈J

w_j²hΓ_jP_K_jS⁻¹y,Γ_jP_K_jyi_Bk²

≤ kX

i∈I

v_i²hΛiP_H_iS⁻¹_Λ x,Λ_iP_H_iS_Λ⁻¹xiAkkX

i∈I

v_i²hΛiP_H_ix,Λ_iP_H_ixiAk kX

j∈J

w_j²hΓjP_K_jS_Γ⁻¹y,Γ_jP_K_jS_Γ⁻¹yiBkkX

j∈J

w²_jhΓjP_K_jy,Γ_jP_K_jyiBk

≤BDkxk²kyk²kX

i∈I

v²_ihΛiPH_iS_Λ⁻¹P_S⁻¹

Λ H_ix,ΛiPH_iS_Λ⁻¹P_S⁻¹

Λ H_ixiAk kX

j∈J

w_j²hΓjPK_jS_Γ⁻¹P_S⁻¹

Γ K_jy,ΓjPK_jS_Γ⁻¹P_S⁻¹

Γ K_jyiBk

=BDkx⊗yk²kX

i,j

(viwj)²h(Λi⊗Γj)PH_i⊗KjS_Λ⊗Γ⁻¹ P_S⁻¹

Λ⊗Γ(Hi⊗Kj)(x⊗y)k,

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hence, 1

BDkx⊗yk²≤ kX

i,j

(viwj)²h(Λi⊗Γj)PH_i⊗KjS_Λ⊗Γ⁻¹ P_S−1

Λ⊗Γ(Hi⊗Kj)(x⊗y)k.

Therefore,θis a g−fusion frame forH⊗K.

Proposition 3.7. For the g−fusion frameθ, frame operator isS_Λ⊗Γ⁻¹ . Proof. We putG= (Λ_i⊗Γ_j)P_H_i_⊗K_jS_Λ⊗Γ⁻¹ , then

G^∗G= ((Λi⊗Γj)PH_i⊗KjS_Λ⊗Γ⁻¹ )^∗(Λi⊗Γj)PH_i⊗KjS_Λ⊗Γ⁻¹

= (S_Λ⁻¹⊗S_Γ⁻¹)(PH_i⊗PK_j)(Λ^∗_i ⊗Γ^∗_j)(Λi⊗Γj)(PH_i⊗PK_j)(S_Λ⁻¹⊗S_Γ⁻¹)

=S⁻¹_Λ P_H_iΛ^∗_iΛ_iP_H_iS_Λ⁻¹⊗S_Γ⁻¹P_K_jΓ^∗_jΓ_jP_K_jS_Γ⁻¹ hence,

P_S−1

Λ⊗Γ(Hi⊗Kj)G^∗GP_S−1

Λ⊗Γ(Hi⊗Kj)

= P_S−1

Λ H_i⊗P_S−1 Γ K_j

G^∗G P_S−1

Λ H_i⊗P_S−1 Γ K_j

=P_S⁻¹

Λ HiS_Λ⁻¹PH_iΛ^∗_iΛiPH_iS_Λ⁻¹P_S⁻¹

Λ Hi⊗P_S⁻¹

Γ KjS_Γ⁻¹PK_jΓ^∗_jΓjPK_jS_Γ⁻¹P_S⁻¹

Γ Kj

= (P_H_iS_Λ⁻¹)^∗Λ^∗_iΛ_iP_H_iS_Λ⁻¹⊗(P_K_jS_Γ⁻¹)^∗Γ^∗_jΓ_jP_K_jS_Γ⁻¹

=S_Λ⁻¹PH_iΛ^∗_iΛiPH_iS_Λ⁻¹⊗S_Γ⁻¹PK_jΓ^∗_jΓjPK_jS_Γ⁻¹. So, for eachx⊗y∈H⊗K, we have

X

i,j

v²_iw²_jP_S⁻¹

Λ⊗Γ(H_i⊗Kj)G^∗GP_S⁻¹

Λ⊗Γ(H_i⊗Kj)(x⊗y)

=X

i,j

v_i²w²_j S_Λ⁻¹PHiΛ^∗_iΛiPHiS⁻¹_Λ ⊗S_Γ⁻¹PKjΓ^∗_jΓjPKjS_Γ⁻¹ (x⊗y)

=

X

i∈I

v²_iS_Λ⁻¹PHiΛ^∗_iΛiPHiS_Λ⁻¹x

⊗

X

j∈J

w_j²S_Γ⁻¹PKjΓ^∗_jΓjPKjS_Γ⁻¹y

=S_Λ⁻¹SΛ(S_Λ⁻¹x)⊗S⁻¹_Γ SΓ(S_Γ⁻¹y)

=S_Λ⁻¹x⊗S_Γ⁻¹y

= (S_Λ⁻¹⊗S_Γ⁻¹)(x⊗y)

=S_Λ⊗Γ⁻¹ (x⊗y),

we conclude that the correspondingg−fusion frame forθ isS_Λ⊗Γ⁻¹ . Theorem 3.8. LetΛ ={Hi,Λi, vi}_i∈I,Λ⁰ ={H_i⁰,Λ⁰_i, v⁰_i}_i∈I beg−fusion Bessel sequences with bounds B, D, respectively in H and Γ = {Kj,Γj, wj}_j∈J, Γ⁰ = {K_j⁰,Γ⁰_j, w⁰_j}_j∈J be g−fusion Bessel sequence with boundsE, F, respectively inK. Suppose(TΛ, T_Λ⁰)and(TΓ, T_Γ⁰)are their

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synthesis operators such that T_Λ⁰T_Λ^∗ = I_H and T_ΓT^∗

Γ⁰ = I_K. Then Λ⊗Γ = {H_i⊗K_j,Λ_i⊗ Γj, viwj}i,j andΛ⁰⊗Γ⁰ ={H_i⁰⊗K_j⁰,Λ⁰_i⊗Γ⁰_j, v⁰_iw⁰_j}i,j areg−fusion frames forH⊗K.

Proof. By theorem3.3, Λ⊗Γ and Λ⁰⊗Γ⁰ areg−fusion Bessel sequences, respectively inH⊗K.

Now, for eachx⊗y∈H⊗K,

kx⊗yk⁴=khx, xiAk²khy, yiBk²

=khT_Λ^∗x, T_Λ^∗0xi_Ak²khT_Γ^∗y, T_Γ^∗0yi_Bk²

≤ kT_Λ^∗xk²kT_Λ^∗0xk²kT_Γ^∗yk²kT_Γ^∗0yk²

≤DFkxk²kyk²kX

i∈I

v_i²hΛiP_H_ix,Λ_iP_H_ixiAkkX

j∈J

w²_jhΓjP_K_jy,Γ_jP_K_jyiBk

=DFkx⊗yk²kX

i,j

v²_iw²_jhΛiPH_ix,ΛiPH_ixiA⊗ hΓjPK_jy,ΓjPK_jyiBk

=DFkx⊗yk²kX

i,j

v²_iw²_jh(Λi⊗Γj)PH_i⊗K_j(x⊗y),(Λi⊗Γj)PH_i⊗K_j(x⊗y)ik,

then, 1

DFkx⊗yk²≤ kX

i,j

v_i²w_j²h(Λi⊗Γj)PH_i⊗K_j(x⊗y),(Λi⊗Γj)PH_i⊗K_j(x⊗y)ik.

Therefore, Λ⊗Γ is ag−fusion frame forH⊗K. Similarly, it can be shown that Λ⁰⊗Γ⁰ is also

ag−fusion frame forH⊗K.

4. Frame operator for a pair ofg−fusion Bessel sequences in tensor product of HilbertC^∗−modules

Definition 4.1. Let Λ⊗Γ ={Hi⊗Kj,Λi⊗Γj, viwj}i,jand Λ⁰⊗Γ⁰ ={H_i⁰⊗K_j⁰,Λ⁰_i⊗Γ⁰_j, v_i⁰w_j⁰}i,j

be twog−fusion Bessel sequences in H⊗K. Then the operatorS:H⊗K→H⊗K defined by for allx⊗y∈H⊗K,

S(x⊗y) =X

i,j

viwjv⁰_iw⁰_jPH_i⊗K_j(Λi⊗Γj)^∗(Λ⁰_i⊗Γ⁰_j)P_H⁰

i⊗K⁰_j(x⊗y)

is called the frame operator for the pair ofg−fusion Bessel sequences Λ⊗Γ and Λ⁰⊗Γ⁰.

Theorem 4.2. Let S_ΛΛ⁰ and S_ΓΓ⁰ be the frame operators for the pair of g−fusion Bessel sequences

Λ ={Hi,Λ_i, v_i}i∈I,Λ⁰ ={H_i⁰,Λ⁰_i, v⁰_i}j∈J

and

Γ ={Kj,Γ_j, w_j}j∈J, Γ⁰ ={K_j⁰,Γ⁰_j, w⁰_j}j∈J

inH andK, respectively. ThenS=S_ΛΛ⁰ ⊗S_ΓΓ⁰.

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