Frames play significant role in signal and image processing, which leads to many applications in differents fields. In this paper we define the dual of ∗-operator frames and we show their propreties obtained in Hilbert A-modules and we establish some results

Bourouihiya, Abdelkrim

Labrigui, H.

Rossafi, M.

Touri, A.

English

NSU Works

Nova Southeastern UniversityNSUWorksMathematics Faculty Articles Department of Mathematics1-11-2019The Duals of *-Operator Frames for End*A(H)Abdelkrim BourouihiyaNova Southeastern University, ab1221@nova.eduM. RossafiUniversity of Ibn Tofail - Kenitra, MoroccoH. LabriguiUniversity of Ibn Tofail - Kenitra, MoroccoA. TouriUniversity of Ibn Tofail - Kenitra, MoroccoFollow this and additional works at: https://nsuworks.nova.edu/math_facarticlesPart of the Mathematics CommonsThis Article is brought to you for free and open access by the Department of Mathematics at NSUWorks. It has been accepted for inclusion inMathematics Faculty Articles by an authorized administrator of NSUWorks. For more information, please contact nsuworks@nova.edu.NSUWorks CitationBourouihiya, Abdelkrim; Rossafi, M.; Labrigui, H.; and Touri, A., "The Duals of *-Operator Frames for End*A(H)" (2019).Mathematics Faculty Articles. 259.https://nsuworks.nova.edu/math_facarticles/259arXiv:1901.03702v1  [math.OA]  11 Jan 2019THE DUALS OF ∗-OPERATOR FRAMES FOR End∗A(H)A. BOUROUIHIYA2, M. ROSSAFI1∗, H. LABRIGUI 1 and A. TOURI1Abstract. Frames play significant role in signal and image processing, whichleads to many applications in differents fields. In this paper we define thedual of ∗-operator frames and we show their propreties obtained in HilbertA-modules and we establish some results.Frame theory is recently an active research area in mathematics, computerscience, and engineering with many exciting applications in a variety of differentfields. They are generalizations of bases in Hilbert spaces. Frames for Hilbertspaces were first introduced in 1952 by Duffin and Schaeffer [5] for study ofnonharmonic Fourier series. They were reintroduced and developed in 1986 byDaubechies, Grossmann and Meyer [4], and popularized from then on. HilbertC∗-modules is a generalization of Hilbert spaces by allowing the inner product totake values in a C∗-algebra rather than in the eld of complex numbers. The aimof this papers is to study the dual of ∗-operator frames.The paper is organized as follows:In section 2, we briefly recall the definitions and basic properties of operatorframe and ∗-operator frame in Hilbert C∗-modules.In section 3, we introduce the dual ∗-operator frame, the ∗-operator frametransform and the ∗-frame operator.In section 4, we investigate tensor product of Hilbert C∗-modules, we showthat tensor product of dual ∗-operator frames for Hilbert C∗-modules H and K,present a dual ∗-operator frames for H⊗K.1. PreliminariesLet I be a countable index set. In this section we briefly recall the definitionsand basic properties of C∗-algebra, Hilbert C∗-modules, frame, ∗-frame in HilbertC∗-modules. For information about frames in Hilbert spaces we refer to [1]. Ourreference for C∗-algebras is [3, 2]. For a C∗-algebra A, an element a ∈ A ispositive (a ≥ 0) if a = a∗ and sp(a) ⊂ R+. A+ denotes the set of positiveelements of A.Definition 1.1. [6] .A family of adjointable operators {Ti}i∈I on a Hilbert A-moduleH over a unitalC∗-algebra is said to be an operator frame for End∗A(H), if there exist positiveDate: Received: 15/09/2018;∗Corresponding author.2010 Mathematics Subject Classification. Primary 42C15; Secondary 46L06.Key words and phrases. ∗-frame, ∗-operator frame, C∗-algebra, Hilbert A-modules.12 A. BOUROUIHIYA, M. ROSSAFI, H. LABRIGUI, A. TOURIconstants A,B > 0 such thatA〈x, x〉A ≤∑i∈I〈Tix, Tix〉A ≤ B〈x, x〉A, ∀x ∈ H. (1.1)The numbers A and B are called lower and upper bound of the operator frame,respectively. If A = B = λ, the operator frame is λ-tight. If A = B = 1, it iscalled a normalized tight operator frame or a Parseval operator frame.Definition 1.2. [6].A family of adjointable operators {Ti}i∈I on a Hilbert A-module H over aunital C∗-algebra is said to be an ∗-operator frame for End∗A(H), if there existstwo strictly nonzero elements A and B in A such thatA〈x, x〉AA∗ ≤∑i∈I〈Tix, Tix〉A ≤ B〈x, x〉AB∗, ∀x ∈ H. (1.2)The elements A and B are called lower and upper bounds of the ∗-operatorframe, respectively. If A = B = λ, the ∗-operator frame is λ-tight. If A =B = 1A, it is called a normalized tight ∗-operator frame or a Parseval ∗-operatorframe. If only upper inequality of hold, then {Ti}i∈i is called an ∗-operator Besselsequence for End∗A(H).If the sum in the middle of (2.1) is convergent in norm, the operator frame iscalled standard. If only upper inequality of (2.1) hold, then {Ti}i∈I is called anoperator Bessel sequence for End∗A(H).2. Dual of ∗-operator Frame for End∗A(H)We begin this section with the following definition.Definition 2.1. .Let {Ti}i∈I ⊂ End∗A(H) be an ∗-operator frame for H. If there exists an∗-operator frame {Λi}i∈I such that x =∑i∈I T∗i Λix for all x ∈ H. then the∗-operator frames {Λi}i∈I is called the duals ∗-operator frames of {Ti}i∈I .Example 2.2. .Let A be a Hilbert A-module over itself, let {fj}j∈J be an ∗-frame for A.We define the adjointable A-module map Λfj : A → A by Λfjf = 〈f, fj〉.Clearly, that {Λfj}j∈J is an ∗-operator frame for A.Theorem 2.3. .Every ∗-operator frame for End∗A(H) has a dual ∗-operator frame.Proof. .Let {Ti}i∈I ⊂ End∗A(H) be an ∗-operator for End∗A(H), with ∗-operator S.We see that {TiS−1}i∈I is an ∗-operator frame.Or, ∀x ∈ H we have :Sx =∑i∈IT ∗i TixthenTHE DUALS OF ∗-OPERATOR FRAMES FOR End∗A(H) 3x =∑i∈IT ∗i TiS−1xhence {TiS−1}i∈I is a dual ∗-operator frame of {Ti}i∈I .It is called the canonique dual ∗-operator frame of {Ti}i∈I . Remark 2.4. .Assume that T = {Ti}i∈I is an ∗-operator frame for End∗A(H) with analyticoperator RT and T˜ = {T˜i}i∈I is a dual ∗-operator frame of T with analyticoperator RT , then for any x ∈ H we have:x =∑i∈IT ∗i T˜ix = R∗TRT˜xthis show that every element of H can be reconstructed with a ∗-operator framefor End∗A(H) and its dual.Theorem 2.5. .Let {Λi}i∈I be an ∗-operator frame for End∗A(H) with ∗-operator frame trans-form θ, the ∗-operator frame S and the canonical dual ∗-operator frames {Λ˜i}i∈J.Let {Ωi}i∈I be an arbitrary dual ∗-operator frame of {Λi}i∈I with the ∗-operatorframe transform η; then the folowing statements are true:(1) θ∗η = I.(2) Ωi = Πiη for all i ∈ I.(3) If η′: H 7−→ l2(H) is any adjointable right inverse of θ∗ then {Πiη′}i∈I isa dual ∗-operator frame of {Λ}i∈I with the operator frames transform η′.(4) The ∗-operator frame SΩ of {Ωi}i∈I is equal to S−1 + η∗(I − θS−1θ∗)η.(5) Every adjointable right inverse η′of θ∗ is the forme :η′= θS−1 + (I − θS−1θ∗)ψ for some adjointable map ψ : H 7−→ l2(H)and vice versa.(6) There exist a ∗-bessel operator {∆j}j∈J ∈ End∗A(H) {∆}i∈I whose ∗-operator frame transform is η and yields is η and yieldsΩj = Λ˜j +∆j −∑k∈J Λ˜jΛ∗k∆k, ∀j ∈ JProof. .(1) For f, g ∈ H we have :〈θ∗ηf, g〉 = 〈ηf, θg〉= 〈∑i∈IΩif,∑i∈IΛig〉=∑i∈I〈Ωif,Λig〉 =∑i∈I〈Λ∗Ωif, g〉= 〈∑i∈IΛ∗Ωif, g〉 = 〈f, g〉then θ∗η = I.(2) The proof is clear from the definition4 A. BOUROUIHIYA, M. ROSSAFI, H. LABRIGUI, A. TOURI(3) Since η′is adjointable, it follows from prop3.1 that {Πiη′}i∈I is a ∗-besselsequence in H.Also, since (η′)∗θ = I; (η′)∗ is surjective, by lemme 2.7, for f ∈ H wehave:||(η′)∗η)−1||−1〈f, f〉 ≤ 〈η′f, η′f〉 =∑i∈I〈piiη′f, piiη′f〉clearly, η′is the pre-frame ∗-operator frame transform {Πiη′}i∈I(4)SΩ = η∗η= η∗θS−1 + η∗η − η∗θS−1= η∗θS−1 + η∗η − η∗θS−1θ∗η= η∗θS−1 + η∗(I − θS−1θ∗)η(5) If η′is such a right inverse of θ, thenθS−1 + (I − θS−1θ∗)η′= θS−1 + η′ − θS−1θ∗η′= θS−1 + η′ − θS−1I = η′(6) Let {∆i}i∈I be an ∗-operator bessel sequence for End∗A(H) with the pre-frame operator η. For i ∈ I, let Ωi = Λ˜i+∆i−∑k∈I Λ˜iΛ∗k∆k Let S and θbe the ∗-frame operator and the preframe operator of {∆i}i∈I , resp. wedefine the linear operator ψ : H 7−→ l2(H) by ψf = (Ωif)i∈I . clearly, ψ isadjointable, for every i ∈ I, we havepiiψ = Ωi= ΛiS−1 +∆i − ΛiS−1∑k∈IΛ∗k∆k= ΛiS−1 + piiη −∑k∈IΛ∗k∆k= piiθS−1 + piiη − piiθS−1θ∗η= pii(θS−1 + η − θS−1θ∗η)thenψ = θS−1 + η − θS−1θ∗ηby parts (3) and (5) of the theorem; {Ωi}i∈I becomes a dual of ∗-operator{Λi}i∈IExample 2.6. .Let A be a Hilbert A-module over itself, let {fj}j∈J ⊂ A.We define the adjointable A-module map Λfj : A → A with Λfj .f = 〈f, fj〉,clearly {fj}j∈J is a ∗-frame in A if and only if {Λfj}j∈J is a ∗-operator frame inA.In the folowing, we study the duals of such ∗-operator frame.(a) Let {gj}j∈J ⊂ A for all f ∈ A :THE DUALS OF ∗-OPERATOR FRAMES FOR End∗A(H) 5∑j∈JΛ∗gjΛfjf =∑j∈J〈f, fj〉gj =∑j∈J〈f, gj〉fj =∑j∈JΛ∗fjΛgjf.Therefore, {gj}j∈J is a dual ∗-frame of {fj}j∈J if and only if {Λgj}j∈J ; is a dual∗-operator of {Λfj}j∈J(b) Let S and SΛ be the ∗-frame operators of {fj}j∈J and {Λfj ,A}j∈J respec-tively.For all f ∈ A we have:∑j∈J〈f, fj〉fj =∑j∈Jff ∗j fj =∑j∈J〈〈f, fj〉, f∗j 〉 =∑j∈JΛ∗fjΛfjf.It follows that S = SΛ(c) It is clearly to see that {hj}j∈J ⊂ A is an ∗-bessel sequence if and only if{Λhj ,A}j∈J is an ∗-bessel operator.(d) for a ∗-bessel sequence {hj}j∈J we definegj = S−1fj + hj −∑k∈J〈S−1fj, fk〉hkthen the sequence {gj}j∈J is a dual ∗-frame of {fj}j∈J .By the last theorem, the sequence {Γj}j∈J is a dual ∗-operator frame of {Λfj}j∈J ,whereΓj = Λ˜fj + Λhj +∑k∈JΛ˜fjΛ∗fkΛhk , ∀j ∈ Jnow we clain that Γj = Λgj6 A. BOUROUIHIYA, M. ROSSAFI, H. LABRIGUI, A. TOURIIn fact, ∀f ∈ A we haveΓjf = Λ˜fjf + Λhjf −∑k∈JΛ˜fjΛ∗fkΛhkf= ΛfjS−1f + Λhjf −∑k∈JΛfjS−1Λ∗fk〈f, hk〉= 〈S−1f, fj〉+ 〈f, hj〉 −∑k∈J〈S−1Λ∗fk〈f, hk〉, fj〉= 〈S−1f, fj〉+ 〈f, hj〉 −∑k∈J〈S−1Λ∗fkΛhkf, fj〉= 〈S−1f, fj〉+ 〈f, hj〉 −∑k∈J〈S−1Λhkffk, fj〉= 〈S−1f, fj〉+ 〈f, hj〉 −∑k∈J〈S−1fh∗kfk, fj〉= 〈f, S−1fj〉+ 〈f, hj〉 −∑k∈J〈fh∗kfk, S−1fj〉= 〈f, S−1fj + hj〉 −∑k∈J〈f, f ∗khkS−1fj〉= 〈f, S−1fj + hj −∑k∈J〈S−1fj , fk〉hk〉= 〈f, gj〉 = Λgjf.therefore, every ∗-operator frame of {Λfj}j∈J has the form :Λ˜fj + Λhj −∑k∈JΛ˜fjΛ∗fkΛhkwhere {hj}j∈J is a ∗-bessel sequence in A.3. Tensor productIn this section, we study the tensor product of the duals ∗-operator frames.Theorem 3.1. .Let H and K are two Hilbert C∗-modules over unitary C∗-Algebras A and Brespectively, let {Λi}i∈I ⊂ End∗A(H) and {Γj}j∈J ⊂ End∗B(K) are an ∗-operatorsframes.If {Λ˜i}i∈I is a dual of {Λi}i∈I and {Γ˜j}j∈J is a dual of {Γj}j∈Jthen {Λ˜i ⊗ Γ˜j}i∈I,j∈J is a dual ∗-operator frame of {Λi ⊗ Γj}i∈I,j∈J.Proof. .THE DUALS OF ∗-OPERATOR FRAMES FOR End∗A(H) 7Let x ∈ H and y ∈ K, we have :∑i∈I,j∈J(Λi ⊗ Γj)∗(Λ˜i ⊗ Γ˜j)(x⊗ y) =∑i∈I,j∈J(Λ∗i ⊗ Γ∗j )(Λ˜ix⊗ Γ˜jy)=∑i∈I,j∈J(Λ∗i Λ˜ix⊗ Γ∗j Γ˜jy)=∑i∈IΛ∗i Λ˜ix⊗∑j∈JΓ∗j Γ˜jy= x⊗ ythen ∑i∈I,j∈J(Λi ⊗ Γj)∗(Λ˜i ⊗ Γ˜j) = Ihence {Λ˜i ⊗ Γ˜j}i∈I,j∈J is a dual ∗-operator frames of {Λi ⊗ Γj}i∈I,j∈J . Corollary 3.2. .Let (Λij)0≤i≤n;j∈J be a family of ∗-operator and (Λ˜ij)0≤i≤n;j∈J its their dual,then (Λ˜0j ⊗ Λ˜1j ⊗ ...⊗ Λ˜nj)j∈J is a dual of (Λ0j ⊗ Λ1j ⊗ ...⊗ Λnj)j∈J .Acknowledgment. The authors would like to thank from the anonymous re-viewers for carefully reading of the manuscript and giving useful comments, whichwill help to improve the paper.References[1] O. Christensen, An Introduction to Frames and Riesz bases, Brikhouser,2016.[2] J.B.Conway, A Course In Operator Theory, AMS, V.21, 2000.[3] F. R. Davidson, C∗-algebra by example, Fields Ins. Monog. 1996.[4] I.Daubechies, A.Grossmann, and Y.Meyer, Painless nonorthogonal expansions, J.Math.Phys.27 (1986), 1271-1283[5] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math.Soc. 72 (1952), 341-366.[6] M. Rossafi and S. Kabbaj, ∗-operator frame for End∗A(H), Arxiv preprint arXiv:1806.03993(2018).1Department of Mathematics, University of Ibn Tofail, B.P. 133, Kenitra,MoroccoE-mail address : rossafimohamed@gmail.com; hlabrigui75@gmail; touri.abdo68@gmail.com2Department of Mathematics, Nova Southeastern University, 3301 CollegeAvenue, Fort Lauderdale, Florida, USAE-mail address : ab1221@nova.edu

The Duals of *-Operator Frames for End*A(H)

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The Duals of -Operator Frames for EndA(H)

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