Controlled continuous g-Frames in Hilbert C ∗ -Modules
aa r X i v : . [ m a t h . OA ] S e p Controlled continuous g-Frames in Hilbert C ∗ -Modules H. LABRIGUI ∗ , A. TOURI , and S. KABBAJ Abstract.
Frame Theory has a great revolution in recent years, this Theoryhave been extended from Hilbert spaces to Hilbert C ∗ -modules. The purposeof this paper is the introduction and the study of the concept of ControlledContinuous g-Frames in Hilbert C ∗ -Modules. Also we give some properties. Introduction and preliminaries
The concept of frames in Hilbert spaces has been introduced by Duffin andSchaeffer [5] in 1952 to study some deep problems in nonharmonic Fourier series,after the fundamental paper [4] by Daubechies, Grossman and Meyer, framestheory began to be widely used, particularly in the more specialized context ofwavelet frames and Gabor frames [7].Hilbert C ∗ -module arose as generalizations of the notion Hilbert space. Thebasic idea was to consider modules over C ∗ -algebras instead of linear spaces andto allow the inner product to take values in the C ∗ -algebras [18].Continuous frames defined by Ali, Antoine and Gazeau [19]. Gabardo and Hanin [8] called these kinds frames, frames associated with measurable spaces. Formore details, the reader can refer to [15], [16] and [17].Theory of frames have been extended from Hilbert spaces to Hilbert C ∗ -modules [9], [11], [12], [13], [14].In the following we briefly recall the definitions and basic properties of C ∗ -algebra, Hilbert A -modules. Our reference for C ∗ -algebras is [3, 6]. For a C ∗ -algebra A if a ∈ A is positive we write a ≥ A + denotes the set of positiveelements of A . Definition 1.1. [3]. Let A be a unital C ∗ -algebra and H be a left A -module,such that the linear structures of A and H are compatible. H is a pre-Hilbert A -module if H is equipped with an A -valued inner product h ., . i A : H × H → A ,such that is sesquilinear, positive definite and respects the module action. In theother words,(i) h x, x i A ≥ x ∈ H and h x, x i A = 0 if and only if x = 0.(ii) h ax + y, z i A = a h x, y i A + h y, z i A for all a ∈ A and x, y, z ∈ H .(iii) h x, y i A = h y, x i ∗A for all x, y ∈ H . Mathematics Subject Classification.
Key words and phrases.
Continuous g-Frames, Controlled continuous g-frames, C ∗ -algebra,Hilbert A -modules. ∗ Corresponding author. ∗ , A. TOURI, S. KABBAJ For x ∈ H , we define || x || = ||h x, x i A || . If H is complete with || . || , it is called aHilbert A -module or a Hilbert C ∗ -module over A . For every a in C ∗ -algebra A ,we have | a | = ( a ∗ a ) and the A -valued norm on H is defined by | x | = h x, x i A forall x ∈ H .Let H and K be two Hilbert A -modules, A map T : H → K is said to beadjointable if there exists a map T ∗ : K → H such that h T x, y i A = h x, T ∗ y i A forall x ∈ H and y ∈ K .We reserve the notation End ∗A ( H , K ) for the set of all adjointable operatorsfrom H to K and End ∗A ( H , H ) is abbreviated to End ∗A ( H ).The following lemmas will be used to prove our mains results Lemma 1.2. [10]. Let H be Hilbert A -module. If T ∈ End ∗A ( H ) , then h T x, T x i ≤ k T k h x, x i ∀ x ∈ H . Lemma 1.3. [1]. Let H and K two Hilbert A -modules and T ∈ End ∗ ( H , K ) .Then the following statements are equivalent: (i) T is surjective. (ii) T ∗ is bounded below with respect to norm, i.e., there is m > such that k T ∗ x k ≥ m k x k for all x ∈ K . (iii) T ∗ is bounded below with respect to the inner product, i.e., there is m ′ > such that h T ∗ x, T ∗ x i ≥ m ′ h x, x i for all x ∈ K . Lemma 1.4. [2]. Let H and K two Hilbert A -modules and T ∈ End ∗ ( H , K ) .Then: (i) If T is injective and T has closed range, then the adjointable map T ∗ T is invertible and k ( T ∗ T ) − k − ≤ T ∗ T ≤ k T k . (ii) If T is surjective, then the adjointable map T T ∗ is invertible and k ( T T ∗ ) − k − ≤ T T ∗ ≤ k T k . Controlled continuous g-Frames in Hilbert C ∗ -Modules Let X be a Banach space, (Ω , µ ) a measure space, and function f : Ω → X a measurable function. Integral of the Banach-valued function f has definedBochner and others. Most properties of this integral are similar to those ofthe integral of real-valued functions. Because every C ∗ -algebra and Hilbert C ∗ -module is a Banach space thus we can use this integral and its properties.Let (Ω , µ ) be a measure space, let U and V be two Hilbert C ∗ -modules, { V w } w ∈ Ω is a sequence of subspaces of V, and End ∗A ( U, V w ) is the collection of all adjointable A -linear maps from U into V w . We define ⊕ w ∈ Ω V w = (cid:26) x = { x w } w ∈ Ω : x w ∈ V w , (cid:13)(cid:13)(cid:13)(cid:13)Z Ω | x w | dµ ( w ) (cid:13)(cid:13)(cid:13)(cid:13) < ∞ (cid:27) . For any x = { x w } w ∈ Ω and y = { y w } w ∈ Ω , if the A -valued inner product is definedby h x, y i = R Ω h x w , y w i dµ ( w ), the norm is defined by k x k = kh x, x ik , the ⊕ w ∈ Ω V w ONTROLLED CONTINUOUS G-FRAMES IN HILBERT C ∗ -MODULES 3 is a Hilbert C ∗ -module. Let GL + ( U ) be the set for all positive bounded linearinvertible operators on U with bounded inverse. Definition 2.1. [20] We call { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } a continuous g-framefor Hilbert C ∗ -module U with respect to { V w : w ∈ Ω } if: • for any x ∈ U , the function ˜ x : Ω → V w defined by ˜ x ( w ) = Λ w x ismeasurable; • there exist two strictly nonzero elements A and B in A such that(2.1) A h x, x i ≤ Z Ω h Λ w x, Λ w x i dµ ( w ) ≤ B h x, x i , ∀ x ∈ U. The elements A and B are called continuous g-frame bounds.If A = B we call this continuous g-frame a continuous tight g-frame, andif A = B = 1 A it is called a continuous Parseval g-frame. If only the right-hand inequality of (2.1) is satisfied, we call { Λ w : w ∈ Ω } a continuous g-Besselsequence for U with respect to { V w : w ∈ Ω } with Bessel bound B .The contnuous g-frame operator S on U is : Sx = Z Ω Λ ∗ ω Λ ω xdµ ( ω )The frame operator S is a bounded, positive, selfadjoint, and invertible (see [20]) Theorem 2.2. [20] Let
Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } , then Λ be a con-tinuous g-frame for U with respect to { V w : w ∈ Ω } if and only if there exist aconstants A and B such that for any x ∈ U : A k x k ≤ k Z Ω h Λ w x, Λ w x i dµ ( w ) k ≤ B k x k Definition 2.3.
Let
C, C ′ ∈ GL + ( U ), we call Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } a ( C − C ′ )-controlled continuous g-frame for Hilbert C ∗ -module U with respect to { V w : w ∈ Ω } if Λ is continuous g-Bessel sequence and there exist two constants A >
B < ∞ such that :(2.2) A h x, x i ≤ Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) ≤ B h x, x i , ∀ x ∈ U.A and B are called the ( C − C ′ )-controlled continuous g-frames bounds.If C ′ = I then we call Λ a C -controlled continuous g-frames for U with respectto { V w : w ∈ Ω } .Let Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } be a continuous g-frames for U withrespect to { V w : w ∈ Ω } .The bounded linear operator T CC ′ : l ( { V w } w ∈ Ω ) → U given by T CC ′ ( { y w } w ∈ Ω ) = Z Ω ( CC ′ ) Λ ∗ ω y ω dµ ( w ) ∀{ y w } w ∈ Ω ∈ l ( { V w } w ∈ Ω )is called the synthesis operator for the ( C − C ′ )-controlled continuous g-frame { Λ w } w ∈ Ω . H. LABRIGUI ∗ , A. TOURI, S. KABBAJ The adjoint operator T ∗ CC ′ : U → l ( { V w } w ∈ Ω ) given by(2.3) T ∗ CC ′ ( x ) = { Λ ω ( C ′ C ) x } ω ∈ Ω ∀ x ∈ U is called the analysis operator for the ( C − C ′ )-controlled continuous g-frame { Λ w w ∈ Ω } .When C and C ′ commute with each other, and commute with the operator Λ ∗ ω Λ ω for each ω ∈ Ω, then the ( C − C ′ )-controlled continuous g-frames operator: S CC ′ : U −→ U is defined as: S CC ′ x = T CC ′ T ∗ CC ′ x = R Ω C ′ Λ ∗ w Λ w Cxdµ ( w )From now on we assume that C and C ′ commute with each other, and commutewith the operator Λ ∗ ω Λ ω for each ω ∈ Ω Proposition 2.4.
The ( C − C ′ ) -controlled continuous g-frames operator S CC ′ isbounded, positive, sefladjoint and invertible.Proof. . We show that S CC ′ is a bounded operator: k S CC ′ k = sup x ∈ U, k x k≤ kh S CC ′ x, x ik = sup x ∈ U, k x k≤ Z Ω C ′ Λ ∗ w Λ w Cxdµ ( w ) k ≤ B From the ( C − C ′ )-controlled continuous g-frames identity (2.2), we have: A h x, x i ≤ h S CC ′ x, x i ≤ B h x, x i so A.Id U ≤ S CC ′ ≤ B.Id U Where Id U is the identity operator in U .We clearly see that S CC ′ is a positive operator.Thus the ( C − C ′ )-controlled continuous g-frames operator S CC ′ is bounded andinvertibleIn other hand we know every positive operator is self adjoint. (cid:3) Theorem 2.5.
Let
Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } is ( C − C ′ ) -controlledcontinuous g-bessel sequence for U , then Λ is a ( C − C ′ ) -controlled continuousg-frames for U with respect to { V w : w ∈ Ω } if and only if there exist a positiveconstants A and B such that : (2.4) A k x k ≤ k Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) k ≤ B k x k ∀ x ∈ U. Proof.
Let { Λ w } w ∈ Ω be a ( C − C ′ )-controlled continuous g-frames for U withrespect to { V w : w ∈ Ω } with bounds A and B .Hence, we have(2.5) A h x, x i ≤ Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) ≤ B h x, x i ∀ x ∈ U. Since 0 ≤ h x, x i , ∀ x ∈ U , then we can take the norme in the left, middle andright termes of the above inequality (2.5).Thus we have: k A h x, x ik ≤ k Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) k ≤ k B h x, x ik ∀ x ∈ U. ONTROLLED CONTINUOUS G-FRAMES IN HILBERT C ∗ -MODULES 5 So, A k x k ≤ k Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) k ≤ B k x k ∀ x ∈ U. Conversely, suppose that (2.4) holds,we have:(2.6) h S CC ′ x, S CC ′ x i = h S CC ′ x, x i = Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w )using (2.6) in (2.5), we get for all x ∈ U : √ A k x k ≤ k S CC ′ x k ≤ √ B k x k by lemma 1.3 ∃ m, M > m h x, x i ≤ h S CC ′ x, S CC ′ x i ≤ M h x, x i Therefore { Λ w : w ∈ Ω } is a ( C − C ′ )-controlled continuous g-frames for U withrespect to { V w : w ∈ Ω } (cid:3) Theorem 2.6.
Let C ∈ GL + ( U ) , the sequence Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } is a continuous g-frame for U with respect to { V w : w ∈ Ω } if and only if Λ is a ( C − C ) -controlled continuous g-frames for U with respect to { V w : w ∈ Ω } Proof.
Suppose that { Λ w : w ∈ Ω } is ( C − C )-controlled continuous g-frameswith bounds A and B , then : A k x k ≤ k Z Ω h Λ w Cx, Λ w Cx i dµ ( w ) k ≤ B k x k ∀ x ∈ U. For any x ∈ U , we have: A k x k = A k CC − x k ≤ A k C k k C − x k ≤ k C k k Z Ω h Λ w CC − x, Λ w CC − x i dµ ( w ) k = k C k k Z Ω h Λ w x, Λ w x i dµ ( w ) k hence,(2.7) A k C k − k x k ≤ k Z Ω h Λ w x, Λ w x i dµ ( w ) k on the other hand : k Z Ω h Λ w x, Λ w x i dµ ( w ) k = k Z Ω h Λ w CC − x, Λ w CC − x i dµ ( w ) k (2.8) k Z Ω h Λ w x, Λ w x i dµ ( w ) k ≤ B k C − x k ≤ B k C − k k x k From (2.7) and (2.8) and theorem2.2 we conclude that { Λ w , w ∈ Ω } is a contin-uous g-frame with bounds A k C k − and B k C − k Conversely, let { Λ w , w ∈ Ω } is a continuous g-frame with bounds E and F , then H. LABRIGUI ∗ , A. TOURI, S. KABBAJ for all x ∈ U we have : E h x, x i ≤ Z Ω h Λ w x, Λ w x i dµ ( w ) ≤ F h x, x i So, for all x ∈ U , Cx ∈ U , and :(2.9) Z Ω h Λ w Cx, Λ w Cx i dµ ( w ) ≤ F h Cx, Cx i ≤ F k C k h x, x i Also, for all x ∈ U , E h x, x i = E h C − Cx, C − Cx i ≤ E k C − k h Cx, Cx i then,(2.10) E h x, x i ≤ k C − k Z Ω h Λ w Cx, Λ w Cx i dµ ( w )From (2.9) and (2.10), we have: E k C − k − h x, x i ≤ Z Ω h Λ w Cx, Λ w Cx i dµ ( w ) ≤ F k C k h x, x i Hence Λ is a ( C − C )-controlled continuous g-frames with bounds E k C − k − and F k C k (cid:3) Proposition 2.7.
Let { Λ w , w ∈ Ω } is a continuous g-frame for U with respectto { V w : w ∈ Ω } and S the continuous g-frame operator associated. Let C, C ′ ∈ GL + ( U ) , then { Λ w , w ∈ Ω } is ( C − C ′ ) -controlled continuous g-framesProof. Let { Λ w , w ∈ Ω } is a continuous g-frame with bounds A and B .by theorem (2.2) we have: A k x k ≤ k Z Ω h Λ w x, Λ w x i dµ ( w ) k ≤ B k x k ∀ x ∈ U (2.11) = ⇒ A k x k ≤ kh Sx, x ik ≤ B k x k ∀ x ∈ U and(2.12) k Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) k = k C kk C ′ kk Z Ω h Λ w x, Λ w x i dµ ( w ) k = k C kk C ′ kkh Sx, x ik From (2.11) and (2.12), we have : A k C kk C ′ kk x k ≤ k Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) k ≤ B k C kk C ′ kk x k ∀ x ∈ U we conclude by theoreme 2.5 that { Λ w , w ∈ Ω } is ( C − C ′ )-controlled continuousg-frames with bounds A k C kk C ′ k and B k C kk C ′ k (cid:3) ONTROLLED CONTINUOUS G-FRAMES IN HILBERT C ∗ -MODULES 7 Theorem 2.8.
Let { Λ w , w ∈ Ω } ⊂ End ∗ A ( U, V ω ) and let C, C ′ ∈ GL + ( U ) so that C, C ′ commute with each other and commute with Λ ∗ ω Λ ω for all ω ∈ Ω . Then thefollowing are equivalent :(1) the sequence { Λ w , w ∈ Ω } is a ( C − C ′ ) -controlled continuous g-Bessel se-quence for U with respect { V ω } ω ∈ Ω with bounds A and B (2) The operator T CC ′ : l ( { V w } w ∈ Ω ) → U given by T CC ′ ( { y w } w ∈ Ω ) = Z w ∈ Ω ( CC ′ ) Λ ∗ ω y ω dµ ( w ) ∀{ y w } w ∈ Ω ∈ l ( { V w } w ∈ Ω ) is well defined and bounded operator with k T CC ′ k ≤ √ B Proof. (1) = ⇒ (2)Let { Λ w , w ∈ Ω } be a ( C − C ′ )-controlled continuous g-Bessel sequence for U with respect { V ω } ω ∈ Ω with bound B .From theorem 2.5 we have :(2.13) k Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) k ≤ B k x k ∀ x ∈ U. For any sequence { y w } w ∈ Ω ∈ l ( { V ω } ω ∈ Ω ) k T CC ′ ( { y w } w ∈ Ω ) k = sup x ∈ U, k x k =1 kh T CC ′ ( { y w } w ∈ Ω ) , x ik = sup x ∈ U, k x k =1 kh Z Ω ( CC ′ ) Λ ∗ ω y ω dµ ( w ) , x ik = sup x ∈ U, k x k =1 k Z Ω h ( CC ′ ) Λ ∗ ω y ω , x i dµ ( w ) k = sup x ∈ U, k x k =1 k Z Ω h y ω , Λ ω ( CC ′ ) x i dµ ( w ) k ≤ sup x ∈ U, k x k =1 k Z Ω h y ω , y ω i dµ ( w ) kk Z Ω h Λ ω ( CC ′ ) x, Λ ω ( CC ′ ) x i dµ ( w ) k = sup x ∈ U, k x k =1 k Z Ω h y ω , y ω i dµ ( w ) kk Z Ω h Λ ω Cx, Λ ω C ′ x i dµ ( w ) k≤ sup x ∈ U, k x k =1 k Z Ω h y ω , y ω i dµ ( w ) k B k x k = B k{ y ω } ω ∈ Ω k Then we have k T CC ′ ( { y w } w ∈ Ω ) k ≤ B k{ y ω } ω ∈ Ω k = ⇒ k T CC ′ k ≤ √ B we conclude the operator T CC ′ is well defined and bounded(2) = ⇒ (1)Let the operator T CC ′ is well defined, bounded and k T CC ′ k ≤ √ B For any x ∈ U and finite subset Ψ ⊂ Ω, we have: Z Ψ h Λ w Cx, Λ w C ′ x i dµ ( w ) = Z Ψ h C ′ Λ ∗ w Λ w Cx, x i dµ ( w ) H. LABRIGUI ∗ , A. TOURI, S. KABBAJ = Z Ψ h ( CC ′ ) Λ ∗ w Λ w ( CC ′ ) x, x i dµ ( w )= h T CC ′ ( { y w } w ∈ Ψ ) , x i≤ k T CC ′ kk ( { y w } w ∈ Ψ ) kk x k Where: y w = Λ w ( CC ′ ) x if ω ∈ Ψ and y w = 0 if ω / ∈ ΨTherefore, Z Ψ h Λ w Cx, Λ w C ′ x i dµ ( w ) ≤ k T CC ′ k ( Z Ψ k Λ w ( CC ′ ) x k dµ ( w )) k x k = k T CC ′ k ( Z Ψ h Λ w Cx, Λ w C ′ x i dµ ( w )) k x k Since Ψ is arbitrary, we have: Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) ≤ k T CC ′ k k x k = ⇒ Z Ω h Λ w Cx, Λ w C ′ x i dµ ( w ) ≤ B k x k as : k T CC ′ k ≤ √ B Therfore { Λ w , w ∈ Ω } is a ( C − C ′ )-controlled continue g-Bessel sequence for U with respect to { V ω } ω ∈ Ω (cid:3) Proposition 2.9.
Let
Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } and Γ = { Γ w ∈ End ∗A ( U, V w ) : w ∈ Ω } be two ( C − C ′ ) -controlled continue g-Bessel sequence for U with respect to { V ω } ω ∈ Ω with bounds E and E respectively. Then the operator L CC ′ : U −→ U given by: (2.14) L CC ′ ( x ) = Z Ω C ′ Γ ∗ w Λ ω Cxdµ ( w ) ∀ x ∈ U is well defined and bounded with k L CC ′ k ≤ √ E E . Also its adjoint operator is L ∗ CC ′ ( g ) = R Ω C ′ Λ ∗ w Γ ω Cxdµ ( w ) Proof. for any x ∈ U and Ψ ⊂ Ω, we have : k Z Ψ C ′ Γ ∗ w Λ ω Cxdµ ( w ) k = sup y ∈ U, k y k =1 kh Z Ψ C ′ Γ ∗ w Λ ω Cxdµ ( w ) , y ik = sup y ∈ U, k y k =1 k Z Ψ h Λ ω Cx, Γ w C ′ y i dµ ( w ) k ≤ sup y ∈ U, k y k =1 k Z Ψ h Λ ω Cx, Λ ω Cx i dµ ( w ) kk Z Ψ h Γ w C ′ y, Γ w C ′ y i dµ ( w ) k≤ k Z Ψ h Λ ω Cx, Λ ω Cx i dµ ( w ) k E ≤ E E k x k since Ψ is arbitrary, R Ψ C ′ Γ ∗ w Λ ω Cxdµ ( w ) converge in U and k L CC ′ k = k Z Ψ C ′ Γ ∗ w Λ ω Cxdµ ( w ) k ≤ p E E ONTROLLED CONTINUOUS G-FRAMES IN HILBERT C ∗ -MODULES 9 In other hand, we have: h L CC ′ x, y i = h Z Ψ C ′ Γ ∗ w Λ ω Cxdµ ( w ) , y i = Z Ψ h C ′ Γ ∗ w Λ ω Cx, y i dµ ( w )= Z Ψ h x, C Λ ∗ w Γ ω C ′ y i dµ ( w )= h x, Z Ψ C Λ ∗ w Γ ω C ′ ydµ ( w ) i Thus L ∗ CC ′ ( g ) = R Ω C ′ Λ ∗ w Γ ω Cxdµ ( w ) (cid:3) Theorem 2.10.
Let
Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } be a ( C − C ′ ) -controlledcontinue g-frames for U with respect to { V ω } ω ∈ Ω and Γ = { Γ w ∈ End ∗A ( U, V w ) : w ∈ Ω } be a ( C − C ′ ) -controlled continue g-Bessel sequence for U with respectto { V ω } ω ∈ Ω . Assume that C and C ′ commute with each other and commute with Γ ∗ w Γ w . If the operator L CC ′ defined in (2.14) is surjective then Γ = { Γ w : w ∈ Ω } is also a ( C − C ′ ) -controlled continue g-frames for U with respect to { V ω } ω ∈ Ω Proof.
Let Λ = { Λ w ∈ End ∗A ( U, V w ) : w ∈ Ω } be a ( C − C ′ )-controlled continueg-frames for U with respect to { V ω } ω ∈ Ω .by theorem 2.8, the operator T CC ′ : l ( { V w } w ∈ Ω ) → U given by T CC ′ ( { y w } w ∈ Ω ) = Z w ∈ Ω ( CC ′ ) Λ ∗ ω y ω dµ ( w ) ∀{ y w } w ∈ Ω ∈ l ( { V w } w ∈ Ω )is well defined and bounded operator.By (2.3) its adjoint operator T ∗ CC ′ : U → l ( { V w } w ∈ Ω ) given by(2.15) T ∗ CC ′ ( x ) = { Λ ω ( C ′ C ) x } ω ∈ Ω Since Γ = { Γ w : w ∈ Ω } is also a ( C − C ′ )-controlled continue g-Bessel sequencefor U with respect to { V ω } ω ∈ Ω .Again by theorem 2.8, the operator K CC ′ : l ( { V w } w ∈ Ω ) → U given by K CC ′ ( { y w } w ∈ Ω ) = Z w ∈ Ω ( CC ′ ) Γ ∗ ω y ω dµ ( w ) ∀{ y w } w ∈ Ω ∈ l ( { V w } w ∈ Ω )is well defined and bounded operator. Again its adjoint operator is given by K ∗ CC ′ ( x ) = { Γ ω ( C ′ C ) x } ω ∈ Ω ∀ x ∈ U Hence for any x ∈ U , the operator defined in (2.14) can be written as : L CC ′ ( x ) = Z Ω C ′ Γ ∗ w Λ ω Cxdµ ( w ) = K CC ′ T ∗ CC ′ x Since L CC ′ is surjective then for any x ∈ U , there exists y ∈ U such that: x = L CC ′ x = K CC ′ T ∗ CC ′ x and T ∗ CC ′ x ∈ l ( { V w } w ∈ Ω )This implies that K CC ′ is surjective. As a result of lemma1.4, we have K ∗ CC ′ isbounded below, that is there exists m > h K ∗ CC ′ x, K ∗ CC ′ x i ≥ m h x, x i ∀ x ∈ U ∗ , A. TOURI, S. KABBAJ = ⇒ h K CC ′ K ∗ CC ′ x, x i ≥ m h x, x i ∀ x ∈ U = ⇒ h Z Ω ( CC ′ ) Γ ∗ w Γ ω ( CC ′ ) xdµ ( ω ) , x i ≥ m h x, x i ∀ x ∈ U = ⇒ Z Ω h Γ w Cx, Γ w C ′ x i dµ ( ω ) ≥ m h x, x i ∀ x ∈ U Hence Γ = { Γ w : w ∈ Ω } is also a ( C − C ′ )-controlled continue g-frames for U with respect to { V ω } ω ∈ Ω (cid:3) References
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