An appropriate representation space for controlled G-frames
Maryam Forughi, Elnaz Osgooei, Asghar Rahimi, Mojgan Javahernia
aa r X i v : . [ m a t h . F A ] M a y AN APPROPRIATE REPRESENTATION SPACE FORCONTROLLED G-FRAMES A MARYAM FORUGHI, B ELNAZ OSGOOEI ∗ , C ASGHAR RAHIMI,AND D MOJGAN JAVAHERNIA
Abstract.
In this paper, motivating the range of operators, we propose anappropriate representation space to introduce synthesis and analysis operatorsof controlled g-frames and discuss the properties of these operators. Especially,we show that the operator obtained by the composition of the synthesis andanalysis operators of two controlled g-Bessel sequence is a trace class operator.Also, we define the canonical controlled g-dual and show that this dual givesrise to expand coefficients with the minimal norm. Finally, we extend someknown equalities and inequalities for controlled g-frames. Introduction and Preliminaries
Frames were first introduced in the context of non-harmonic Fourier series byDuffin and Schaeffer [5]. During the last 20s the theory of frames has been devel-oped rapidly and because of the abundant use of frames in engineering and appliedsciences, many generalization of frames have come into play.G-frames that include the concept of ordinary frames have been introduced by Sun[15] and improved by many authors [6, 8, 10, 14]. Controlled frames have been im-proved recently to improve the numerical efficiency of interactive algorithms thatinverts the frame operator [2]. Following that, controlled frames have been gener-alized to another kinds of frames [6, 7, 9, 13, 14, 12, 8].In this paper, motivating the concept of g-frames and controlled frames we de-fine controlled g-frames. In Section 2, imagined the range of an operator, a newrepresentation space is introduced such that the synthesis and analysis operatorscould be defined. In Section 3, controlled g- dual frames and canonical controlledg-dual frames are introduced and shown that canonical g-dual gives rise to expandcoefficients with the minimal norm. Finally, some equalities and inequalities arepresented for controlled g-frames and especially for their operators in Section 4.Throughout this paper, H is a separable Hilbert space, { H i } i ∈ I is the collectionof Hilbert spaces, B ( H, K ) is the family of all linear bounded operators from H into K and GL ( H ) is the set of all bounded linear operators which have boundedinverses.At first, we collect some definitions and basic results that are needed in the paper. Lemma 1.1. ( [11] ) Let u ∈ B ( H ) be a self-adjoint operator and v := au + bu + c where a, b, c ∈ R . (i) If a > , then inf k f k =1 h vf, f i ≥ ac − b a . Corresponding author ∗ Mathematics Subject Classification.
Primary 42C15; Secondary 46C99, 41A58.
Key words and phrases.
Controlled g-frame, controlled g-dual frame, trace class operator. (ii) If a < , then sup k f k =1 h vf, f i ≤ ac − b a . Lemma 1.2. ( [1] ) If u, v are operators on H satisfying u + v = id H , then u − v = u − v . If an operator u has closed range, then there exists a right-inverse operator u † (pseudo-inverse of u ) in the following senses (see [4]). Lemma 1.3.
Let u ∈ B ( K, H ) be a bounded operator with closed range R u . Thenthere exists a bounded operator u † ∈ B ( H, K ) for which uu † x = x, x ∈ R ( u ) . Definition 1.4. ( g-frame ) A family Λ := { Λ i ∈ B ( H, H i ) } i ∈ I is called a g-framefor H with respect to { H i } i ∈ I , if there exist 0 < A ≤ B < ∞ such that(1.1) A k f k ≤ X i ∈ I k Λ i f k ≤ B k f k , f ∈ H. If only the second inequality in (1.1) satisfy, then we say that { Λ i } i ∈ I is a g-Besselsequence with upper bound B .If Λ is a g-Bessel sequence, then the synthesis and analysis operators are definedby T Λ :( X i ∈ I ⊕ H i ) ℓ → H, T ∗ Λ : H → ( X i ∈ I ⊕ H i ) ℓ ,T Λ ( { f i } i ∈ I ) = X i ∈ I Λ ∗ i ( f i ) , T ∗ Λ ( f ) = { Λ i f } i ∈ I , where ( X i ∈ I ⊕ H i ) ℓ = (cid:8) { f i } i ∈ I | f i ∈ H i , X i ∈ I k f i k < ∞ (cid:9) , and, the g-frame operator is given by S Λ f = T Λ T ∗ Λ f = X i ∈ I Λ ∗ i Λ i f, f ∈ H, which is positive, self-adjoint and invertible (see [15]).2. Controlled g-frames and their operators
Controlled frames for spherical wavelets were introduced in [3] to get a numer-ically more efficient approximation algorithm. In this section by extending theconcept of controlled frames and g-frames, we define the concept of controlled g-frames and construct an appropriate representation space to organize the synthesisand analysis operators.
Definition 2.1. [2] Let C ∈ GL ( H ). We say that F := { f i } i ∈ I is a C -controlledframe for H if there exist 0 < A C ≤ B C < ∞ such that for each f ∈ H (2.1) A C k f k ≤ X i ∈ I h f, f i ih Cf i , f i ≤ B C k f k . N APPROPRIATE REPRESENTATION SPACE FOR CONTROLLED G-FRAMES 3
Definition 2.2.
Let
C, C ′ ∈ GL ( H ). We say that Λ := { Λ i ∈ B ( H, H i ) } i ∈ I is a( C, C ′ )-controlled g-frame for H if there exist 0 < A CC ′ ≤ B CC ′ < ∞ such that foreach f ∈ H (2.2) A CC ′ k f k ≤ X i ∈ I h Λ i C ′ f, Λ i Cf i ≤ B CC ′ k f k . For simplicity, we use a notation CC ′ instead of ( C, C ′ ). We call Λ a Parseval CC ′ -controlled g-frame if A CC ′ = B CC ′ = 1. When the right hand inequality of(2.2) holds, then Λ is called a CC ′ -controlled g-Bessel sequence for H with bound B C .If Λ is a CC ′ -controlled g-frame for H and C ∗ Λ ∗ i Λ i C ′ is positive for each i ∈ I ,then we have A CC ′ k f k ≤ X i ∈ I k ( C ∗ Λ ∗ i Λ i C ′ ) f k ≤ B CC ′ k f k , f ∈ H. Consider a proper representation space by K := (cid:8) { ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I : f ∈ H (cid:9) ⊂ ( X i ∈ I ⊕ H i ) ℓ . Since K is a closed subspace of ( P i ∈ I ⊕ H i ) ℓ , we can define the synthesis andanalysis operators of CC ′ -controlled g-frames by T CC ′ : K −→ H,T CC ′ ( { ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I ) = X i ∈ I C ∗ Λ ∗ i Λ i C ′ f and T ∗ CC ′ : H −→ K,T ∗ CC ′ ( f ) = { ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I . Thus, the CC ′ -controlled g-frame operator is given by S CC ′ f = T CC ′ T ∗ CC ′ f = X i ∈ I C ∗ Λ ∗ i Λ i C ′ f, f ∈ H. So, h S CC ′ f, f i = X i ∈ I h Λ i C ′ f, Λ i Cf i , f ∈ H, and A CC ′ Id H ≤ S CC ′ ≤ B CC ′ Id H . Therefore, S CC ′ is a positive, self-adjoint and invertible operator (see [14] and [9]).Thus, since f = S CC ′ S − CC ′ f = S − CC ′ S CC ′ f , we have(2.3) f = X i ∈ I C ∗ Λ ∗ i Λ i C ′ S − CC ′ f = X i ∈ I S − CC ′ C ∗ Λ ∗ i Λ i C ′ f, f ∈ H. Remark . We introduce a Parseval CC ′ -controlled g-frame for H by the CC ′ -controlled g-frame operator. Suppose that Λ is a CC ′ -controlled g-frame for H .Since S CC ′ (or S − CC ′ ) is positive in B ( H ) and B ( H ) is a C ∗ -algebra, then there M. FORUGHI, E. OSGOOEI, A. RAHIMI, AND M. JAVAHERNIA exists a unique positive square root S CC ′ (or S − CC ′ ) which commutes with S CC ′ and S − CC ′ . Therefore, for any f ∈ H we can write f = S − CC ′ S CC ′ S − CC ′ f = X i ∈ I S − CC ′ C ∗ Λ ∗ i Λ i C ′ S − CC ′ f. Now, assume that S − CC ′ commutes with C, C ′ . Then we get k f k = h f, f i = X i ∈ I h Λ i S − CC ′ C ′ f, Λ i S − CC ′ Cf i . Hence, { Λ i S − CC ′ } i ∈ I is a Parseval CC ′ -controlled g-frame for H . Theorem 2.4.
A sequence Λ is a CC ′ -controlled g-Bessel sequence for H withbound B CC ′ if and only if the operator T CC ′ : K −→ H,T CC ′ ( { ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I ) = X i ∈ I C ∗ Λ ∗ i Λ i C ′ f is a well-defined and bounded operator with k T CC ′ k ≤ √ B CC ′ .Proof. We only need to prove the sufficient condition. Let T CC ′ be a well-definedand bounded operator with k T CC ′ k ≤ √ B CC ′ . For each f ∈ H we have X i ∈ I h Λ i C ′ f, Λ i Cf i = X i ∈ I h C ∗ Λ ∗ i Λ i C ′ f, f i = (cid:10) T CC ′ ( { ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I ) , f (cid:11) ≤ k T CC ′ kk{ ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I kk f k . But k{ ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I k = X i ∈ I h Λ i C ′ f, Λ i Cf i . It follows that X i ∈ I h Λ i C ′ f, Λ i Cf i ≤ B CC ′ k f k , and this means that Λ is a CC ′ -controlled g-Bessel sequence. (cid:3) Theorem 2.5.
A sequence Λ is a CC ′ -controlled g-frame for H if and only if T CC ′ : ( { ( C ∗ Λ ∗ i Λ i C ′ ) f } i ∈ I ) X i ∈ I C ∗ Λ ∗ i Λ i C ′ f is a well-defined and surjective operator.Proof. First, suppose that Λ is a CC ′ -controlled g-frame for H . Since, S CC ′ is asurjective operator, so T CC ′ . For the opposite implication, by Theorem 2.4, T CC ′ isa well-defined and bounded operator. So, Λ is a CC ′ -controlled g-Bessel sequence. N APPROPRIATE REPRESENTATION SPACE FOR CONTROLLED G-FRAMES 5
Now, for each f ∈ H , we have f = T CC ′ T † CC ′ f . Hence, k f k = |h f, f i| = |h T CC ′ T † CC ′ f, f i| = |h T † CC ′ f, T ∗ CC ′ f i| ≤ k T † CC ′ k k f k X i ∈ I h Λ i C ′ f, Λ i Cf i . We conclude that 1 k T † CC ′ k k f k ≤ X i ∈ I h Λ i C ′ f, Λ i Cf i , f ∈ H. (cid:3) Theorem 2.6.
Let
Λ = { Λ i ∈ B ( H, H i ) } i ∈ I and Θ := { Θ i ∈ B ( H, H i ) } i ∈ I be two CC ′ -controlled g-Bessel sequence for H with bounds B and B , respectively. If T Λ and T Θ are their synthesis operators such that T Λ T ∗ Θ = Id H , then both Λ and Θ are CC ′ -controlled g-frames for H .Proof. For each f ∈ H , we have k f k = h f, f i = h T ∗ Λ f, T ∗ Θ f i ≤ k T ∗ Λ f k k T ∗ Θ f k = (cid:0) X i ∈ I h Λ i C ′ f, Λ i Cf i (cid:1)(cid:0) X i ∈ I h Θ i C ′ f, Θ i Cf i (cid:1) ≤ (cid:0) X i ∈ I h Λ i C ′ f, Λ i Cf i (cid:1) B k f k . Hence, B − k f k ≤ (cid:0) X i ∈ I h Λ i C ′ f, Λ i Cf i (cid:1) , and Λ is a CC ′ -controlled g-frame. Similarly, Θ is a CC ′ -controlled g-frame withlower bound B − . (cid:3) Theorem 2.7.
Let
Λ = { Λ i ∈ B ( H, H i ) } i ∈ I and Θ := { Θ i ∈ B ( H, H i ) } i ∈ I be two CC ′ -controlled g-Bessel sequence for H with bounds B and B , respectively where | I | < ∞ . If Φ := T Λ T ∗ Θ , then Φ is a trace class operator.Proof. Suppose that Φ = u | Φ | is the polar decomposition of Φ where u ∈ B ( H ) isa partial isometry. So, | Φ | = u ∗ Φ. Let { e i } i ∈ I be an orthonormal basis for H . We M. FORUGHI, E. OSGOOEI, A. RAHIMI, AND M. JAVAHERNIA have tr( | Φ | ) = X i ∈ I h| Φ | e i , e i i = X i ∈ I h T ∗ Θ e i , T ∗ Λ ue i i = X i ∈ I h{ ( C ∗ Θ ∗ j Θ j C ′ ) e i } j ∈ I , { ( C ∗ Λ ∗ j Λ j C ′ ) ue i } j ∈ I i≤ X i ∈ I X j ∈ I k ( C ∗ Θ ∗ j Θ j C ′ ) e i kk ( C ∗ Λ ∗ j Λ j C ′ ) ue i k≤ X i ∈ I (cid:0) X j ∈ I k ( C ∗ Θ ∗ j Θ j C ′ ) e i k (cid:1) (cid:0) X j ∈ I k ( C ∗ Λ ∗ j Λ j C ′ ) ue i k (cid:1) ≤ X i ∈ I p B B k ue i k < ∞ . (cid:3) Controlled g-dual frames
In this section by considering that C = C ′ and Λ is a C -controlled g-frame for H , we define a canonical controlled g-dual and show that this canonical dual is ag-frame and gives rise to expand coefficients with the minimal norm. Definition 3.1.
Suppose that Λ = { Λ i ∈ B ( H, H i ) } i ∈ I and e Λ := { e Λ i ∈ B ( H, H i ) } i ∈ I are two CC ′ -controlled g-Bessel sequence for H with synthesis operators T Λ and T e Λ , respectively. We say that e Λ is a CC ′ -controlled g-dual of Λ if T Λ T ∗ e Λ = Id H . In this case Λ , e Λ are said CC ′ -controlled g-dual pair also.The proof of the following is straightforward. Proposition 3.2. If Λ , e Λ are CC ′ -controlled g-dual pair, then the following state-ments are equivalent: (i) T Λ T ∗ e Λ = Id H ; (ii) T e Λ T ∗ Λ = Id H ; (iii) h f, g i = h T ∗ e Λ f, T ∗ Λ g i , f, g ∈ H .Also, for every f ∈ H , we have (3.1) f = X i ∈ I ( C ∗ Λ ∗ i Λ i C ′ ) ( C ∗ e Λ ∗ i e Λ i C ′ ) f. Now, we want to present the canonical controlled g-dual by (2.3) in the case that C = C ′ and Λ is a C -controlled g-frame for H . Let Γ i := Λ i CS − C . Therefore, foreach f ∈ H (3.2) f = X i ∈ I C ∗ Λ ∗ i Γ i f = X i ∈ I Γ ∗ i Λ i Cf.
N APPROPRIATE REPRESENTATION SPACE FOR CONTROLLED G-FRAMES 7
We show that Γ := { Γ i } i ∈ I is a g-frame for H . Let f ∈ H and A C , B C be the framebounds of Λ. Then X i ∈ I k Γ i f k = X i ∈ I h Λ i CS − C f, Λ i CS − C f i = X i ∈ I h C ∗ Λ ∗ i Λ i CS − C f, S − C f i = h f, S − C f i≤ A C k f k . On the other hand, k f k = h f, f i = h X i ∈ I Γ ∗ i Λ i Cf, f i = h X i ∈ I Λ i Cf, Γ i f i ≤ (cid:0) X i ∈ I k Λ i Cf k (cid:1)(cid:0) X i ∈ I k Γ i f k (cid:1) ≤ B C k f k (cid:0) X i ∈ I k Γ i f k (cid:1) . Finally, we conclude that1 B C k f k ≤ X i ∈ I k Γ i f k ≤ A C k f k . The following theorem shows that the canonical controlled g-dual gives rise toexpand coefficients with the minimal norm.
Theorem 3.3.
Let
Λ = { Λ i ∈ B ( H, H i ) } i ∈ I be a C -controlled g-frame for H and Γ i = Λ i CS − C . If f has a representation f = P i ∈ I C ∗ Λ ∗ i g i , for some g i ∈ H i . Thenwe have X i ∈ I k g i k = X i ∈ I k Γ i f k + X i ∈ I k g i − Γ i f k , f ∈ H Proof.
Assume that f ∈ H . We get by (3.2) X i ∈ I k Γ i f k = X i ∈ I h Γ i f, Λ i CS − C f i = X i ∈ I h C ∗ Λ ∗ i Γ i f, S − C f i = X i ∈ I h C ∗ Λ ∗ i g i , S − C f i = X i ∈ I h g i , Λ i CS − C f i = X i ∈ I h g i , Γ i f i . Therefore, Im (cid:16) P i ∈ I h g i , Γ i f i (cid:17) = 0 and the conclusion follows. (cid:3) M. FORUGHI, E. OSGOOEI, A. RAHIMI, AND M. JAVAHERNIA Some equalities and inequalities
In this section, we extend some known equalities and inequalities for controlledg-frames. Assume that Λ , e Λ are CC ′ -controlled g-dual pair and J ⊂ I . We define S J f := X i ∈ J ( C ∗ Λ ∗ i Λ i C ′ ) ( C ∗ e Λ ∗ i e Λ i C ′ ) f, f ∈ H. It is clear that S J ∈ B ( H ) and S J + S J c = Id H where J c is the complement of J .Indeed, if B and B are the bounds of Λ and e Λ respectively, then k S J f k = (cid:16) sup k g k =1 |h S J f, g i| (cid:17) ≤ (cid:16) sup k g k =1 X i ∈ J (cid:12)(cid:12)(cid:10) ( C ∗ Λ ∗ i Λ i C ′ ) ( C ∗ e Λ ∗ i e Λ i C ′ ) f, g (cid:11)(cid:12)(cid:12)(cid:17) ≤ (cid:16) X i ∈ I k ( C ∗ Λ ∗ i Λ i C ′ ) f k (cid:17)(cid:16) sup k g k =1 X i ∈ I k ( C ∗ e Λ ∗ i e Λ i C ′ ) g k (cid:17) ≤ B B k f k . So, S J is bounded. Theorem 4.1. If f ∈ H then, X i ∈ J h ( C ∗ e Λ ∗ i e Λ i C ′ ) f, ( C ∗ Λ ∗ i Λ i C ′ ) f i − k S J f k = X i ∈ J c h ( C ∗ e Λ ∗ i e Λ i C ′ ) f, ( C ∗ Λ ∗ i Λ i C ′ ) f i − k S J c f k Proof.
Let f ∈ H . We obtain X i ∈ J h ( C ∗ e Λ ∗ i e Λ i C ′ ) f, ( C ∗ Λ ∗ i Λ i C ′ ) f i − k S J f k = h S J f, f i − k S J f k = h S J f, f i − h S ∗ J S J f, f i = h ( id H − S J ) ∗ S J f, f i = h S ∗ J c ( id H − S J c ) f, f i = h S ∗ J c f, f i − h S ∗ J c S J c f, f i = h f, S J c f i − h S J c f, S J c f i . Now, the proof is completed. (cid:3)
Corollary 4.2. If Λ is a CC ′ -controlled Parseval g-frame for H , then X i ∈ J k ( C ∗ Λ ∗ i Λ i C ′ ) f k − k X i ∈ J ( C ∗ Λ ∗ i Λ i C ′ ) f k = X i ∈ J c k ( C ∗ Λ ∗ i Λ i C ′ ) f k − k X i ∈ J c ( C ∗ Λ ∗ i Λ i C ′ ) f k . Moreover, X i ∈ J k ( C ∗ Λ ∗ i Λ i C ′ ) f k + k X i ∈ J c ( C ∗ Λ ∗ i Λ i C ′ ) f k ≥ k f k . N APPROPRIATE REPRESENTATION SPACE FOR CONTROLLED G-FRAMES 9
Proof. If f ∈ H , we obtain X i ∈ J k ( C ∗ Λ ∗ i Λ i C ′ ) f k + k S J c f k = (cid:10) ( S J + S J c ) f, f (cid:11) = (cid:10) ( S J + id H − S J + S J ) f, f (cid:11) = h ( id H − S J + S J ) f, f i . Now, by Lemma 1.1 for a = 1, b = − c = 1 the inequality holds. (cid:3) Corollary 4.3. If Λ is a CC ′ -controlled Parseval g-frame for H , then ≤ S J − S J ≤ Id H . Proof.
We have S J S J c = S J c S J . Then 0 ≤ S J S J c = S J − S J . Also, by Lemma 1.1,we get S J − S J ≤ id H . (cid:3) Theorem 4.4.
Let Λ be a CC ′ -controlled g-frame with CC ′ -controlled g-frameoperator S CC ′ . Suppose that S − CC ′ commutes with C, C ′ . Then for each f ∈ H , X i ∈ J k S − CC ′ C ∗ Λ ∗ i Λ i C ′ f k + k S − CC ′ S J c f k = X i ∈ J c k S − CC ′ C ∗ Λ ∗ i Λ i C ′ f k + k S − CC ′ S J f k . Proof.
Via Remark 2.3 and Corollary 4.2, if Θ i := Λ i S − CC ′ , then X i ∈ J ( C ∗ Θ ∗ i Θ i C ′ ) f = X i ∈ J ( C ∗ S − CC ′ Λ ∗ i Λ i S − CC ′ C ′ ) f = X i ∈ J ( S − CC ′ C ∗ Λ ∗ i Λ i C ′ S − CC ′ ) f = S − CC ′ S J S − CC ′ f, and also X i ∈ J k ( C ∗ Θ ∗ i Θ i C ′ ) f k − k X i ∈ J ( C ∗ Θ ∗ i Θ i C ′ ) f k == X i ∈ J c k ( C ∗ Θ ∗ i Θ i C ′ ) f k − k X i ∈ J c ( C ∗ Θ ∗ i Θ i C ′ ) f k . Now, by replacing S CC ′ f instead of f in above formulas, the proof is evident. (cid:3) Corollary 4.5.
Let Λ be a CC ′ -controlled g-frame with CC ′ -controlled g-frameoperator S CC ′ . Suppose that S − CC ′ commutes with C, C ′ . Then ≤ S J − S J S − CC ′ S J ≤ S CC ′ . Proof.
In the proof of Theorem 4.4, we showed that X i ∈ J ( C ∗ Θ ∗ i Θ i C ′ ) f = S − CC ′ S J S − CC ′ f. By Corollary 4.3 we get0 ≤ X i ∈ J ( C ∗ Θ ∗ i Θ i C ′ ) f − (cid:0) X i ∈ J ( C ∗ Θ ∗ i Θ i C ′ ) f (cid:1) ≤ Id H . So, 0 ≤ S − CC ′ ( S J − S J S − CC ′ S J ) S − CC ′ ≤ S CC ′ , and it completes the proof. (cid:3) Corollary 4.6.
Let Λ be a CC ′ -controlled g-frame with CC ′ -controlled g-frameoperator S CC ′ . Suppose that S − CC ′ commutes C, C ′ . Then for each f ∈ H , X i ∈ J k S − CC ′ C ∗ Λ ∗ i Λ i C ′ f k + k S − CC ′ S J c f k ≥ k S − CC ′ k − k f k . Proof.
Applying Theorem 4.4 and Corollary 4.2, we obtain X i ∈ J k S − CC ′ C ∗ Λ ∗ i Λ i C ′ f k + k S − CC ′ S J c f k == X i ∈ J k ( C ∗ Θ ∗ i Θ i C ′ ) S CC ′ f k − k X i ∈ J ( C ∗ Θ ∗ i Θ i C ′ ) S CC ′ f k ≥ k S CC ′ f k = 34 h S CC ′ f, f i≥ k S − CC ′ k − k f k . (cid:3) Theorem 4.7.
Let Λ be a Parseval CC ′ -controlled g-frame for H . Then (i) 0 ≤ S J − S J ≤ id H . (ii) 12 id H ≤ S J + S J c ≤ id H .Proof. (i). Since S J + S J c = id H , then S J S J c + S J c = S J c . Thus S J S J c = S J c − S J c = S J c ( id H − S J c ) = S J c S J . But, Λ is a Parseval CC ′ -controlled g-frame, so 0 ≤ S J S J c = S J − S J . On the otherhand, by Lemma 1.2, we get S J − S J ≤ id H . (ii). Since S J S J c = S J c S J , so by Lemma 1.2 S J + S J c = id H − S J S J c = 2 S J − S J + id H ≥ id H . and we get the right inequality by Lemma 1.2 and 0 ≤ S J − S J , S J + S J c ≤ id H + 2 S J − S J ≤ id H . (cid:3) N APPROPRIATE REPRESENTATION SPACE FOR CONTROLLED G-FRAMES 11
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E-mail address : [email protected] b Faculty of Science, Urmia University of Technology, Urmia, Iran
E-mail address : [email protected] c Department of Mathematics, University of Maragheh, Iran,
E-mail address : [email protected] d Islamic Azad University of Shabester, Tabriz-Shabestar, Iran
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