Some Properties of Controlled K-g-Frames in Hilbert C∗-Modules

Frames for Hilbert spaces were introduced by Duffin and Schaefer [2] in 1952 to study some deep problems in nonharmonic Fourier series by abstracting the fundamental notion of Gabor [4] for signal processing. Many generalizations of the concept of frame have been defined in Hilbert C∗-modules [3, 5, 6, 9, 11–15]. Controlled frames in Hilbert spaces have been introduced by P. Balazs [1] to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Rashidi and Rahimi [8] are introduced the concept of Controlled frames in Hilbert C∗−modules. Let A be a unital C∗−algebra, let I be countable index set. Throughout this paper H and L are countably generated Hilbert A−modules and {Hi}i∈I is a sequence of submodules of L. For each i ∈ I, End∗ A(H,Hi) is the collection of all adjointable A−linear maps from H to Hi , and End∗ A(H,H)


Introduction and Preliminaires
Frames for Hilbert spaces were introduced by Duffin and Schaefer [2] in 1952 to study some deep problems in nonharmonic Fourier series by abstracting the fundamental notion of Gabor [4] for signal processing.
Controlled frames in Hilbert spaces have been introduced by P. Balazs [1] to improve the numerical efficiency of iterative algorithms for inverting the frame operator.
Rashidi and Rahimi [8] are introduced the concept of Controlled frames in Hilbert C * −modules.
Let A be a unital C * −algebra, let I be countable index set. Throughout this paper H and L are countably generated Hilbert A−modules and {H i } i∈I is a sequence of submodules of L. (ii) ax + y , z A = a x, z A + y , z A for all a ∈ A and x, y , z ∈ H.
(iii) x, y A = y , x * A for all x, y ∈ H. For x ∈ H we define x = x, x A 1 2 . If H is complete with . , 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) 1 2 and the A-valued norm on H is defined by |x| = (x * x) 1 2 for x ∈ H. Let H and K be tow Hilbert A modules, A map T : H → K is said to be adjointable if there exists a map T * : K → H such that T x, y A = x, T * y A for all y ∈ K and x ∈ H. Lemma 1.1. [18] Suppose that H 1 and H 2 two Hilbert A-Modules H and H). Then the following assertions are equivalent: There exists a mapping U ∈ End * A (H 1 , H 2 ) such that L 1 = L 2 U. Moreover, if above conditions are valid, then there exists a unique operator U such that . If an operator U has a closed range, then there exists a right-inverse operator U † , (pseudo-inverse of U) in the following sense.
be a bounded operator with closed range R(U). Then there exists a bounded operator U † ∈ End * A (H 2 , H 1 ) for which [10] Let H and K two Hilbert A-module and T ∈ End * A (H, K). Then, the following assertions are equivalent: (i) The operator T is bounded and A-linear, (1.1)

Some Properties of Controlled K-g-Frames
Now, we define controlled K-g-Frames in Hilbert C * -modules.
If the right hand of (2.1) holds, Λ is called a (C, C )-controlled K − g−Bessel sequence in Hilbert We call Λ a Parseval C, C -controlled K-g-frame if For simplicity, we will use a notation CC instead of C, C .
If Λ is a CC -controlled g-frame on Hilbert A-module H, and C * Λ * i Λ i C is positive for all i ∈ I, then for each f ∈ H,

Now, let
It is easy to check that R is a closed subspace of ( i∈I ⊕H) 2 . Now, we can define the synthesis and analysis operators of the CC -controlled g-frames as T CC : R → H, and T * CC : H → R, Thus, the CC -controlled g-frame operator is given by S CC is positive, bounded, invertible and self-adjoint. Moreover Proof. We only need to prove the sufficient condition. Let T CC be a well-defined and bounded operator Hence,

It follows that
i∈I and this means that Λ is a CC -controlled g-Bessel sequence.

in Hilbert
A-module if and only if the operator is well defined, bounded and surjective.
Proof. Suppose that Λ is a CC -controlled g-frame in Hilbert A-module. Since, S CC is surjective operator, so T CC . For the opposite implication, by Lemma 2.1; T CC is a well-defined and bounded We conclude that Proof. Assume that Acc and Bcc are the frame bounds of Λ. Hence, Since ker K * = (R(K)) ⊥ and K has a dense range, K * injective. Then from (2.2), for each i ∈ I, we Then, we have S CC Acc KK * , so by Lemma 1.1, there exists an operator U ∈ End * A (H, R) such that Now, we can obtain optimal frame bounds of Λ by the operator U. Indeed, it is obvious that By Lemma 1.1, the equation (2.3) has a unique solution as U 0 such that

Now, we have
In the following, we consider some proper relations between the operators U, K ∈ End * A (H) and C, C ∈ GL + (H) and investigate the cases that {Λ i U} i∈I , {Λ i U * } i∈I can also CC -controlled K-gframe. Next, by putting connections between the operators S Λ , K, C and C , we reach to necessary and sufficient conditions that {Λ i } i∈I can be a Parseval CC -controlled K-g-frames.
We have Assume that K has a closed range and U ∈ End * A (H) such that R(U * ) ⊂ R(K) Also suppose that U * commutes with C and C . Then {Λ i U * } i∈I is a CC -controlled K-g-frame for R(U) if and only if there exists δ > 0 such that for each f ∈ R(U), Therefore, if A CC is a lower frame bound of Λ, we have For the upper bound, it is clear that and Bcc U 2 . Theorem 2.3. Let Λ be a CC -controlled K-g-frame in Hilbert A-module H and the operator K has a dense rang. Assume that U ∈ End * A (H) has a closed range and U and U * commute with C and C . If {Λ i U * } i∈I and {Λ i U} i∈I are CC -controlled K-g-frame in Hilbert A-module H, then U is invertible.
Proof. Suppose that {Λ i U * } i∈I is a CC -controlled K-g-frame in Hilbert A module H with a lower frame bound A 1 , and B 1 . Then for each f ∈ H,

We have
hence, Since K has a dense range, K * is injective. Moreover, R(U) = (ker U * ) ⊥ = H so U is surjective.
Suppose that {Λ i U * } i∈I is a (CC )-controlled K-g-frame in Hilbert A module H with a lower frame bound A 2 and B 2 . Then, for each f ∈ H, Therefore U is injective, since ker U ⊆ ker K * . Thus, U is an invertible operator.
Theorem 2.4. Let Λ be a CC -controlled K-g-frame in Hilbert A-module H and U ∈ End * A (H) be a co-isometry (i.e. UU * = Id H ) such that UK = KU and U * commutes with C and C . Then Proof. Suppose Λ be a CC -controlled K-g-frame in Hilbert A-module H with a lower frame bound A CC . and B CC for each f ∈ H, we have So, {Λ i U * } i∈I is a CC -controlled g-Bessel sequence. For the lower bound, we can write Suppose that T Λ,C,C and T ,CC are their synthesis operators such that T ,CC T * Λ,C,C = K * . Then Λ and are CC -controlled K and K * -g-frames, respectively. Proof. So, This that Λ is a CC -controlled K-g-frame in Hilbert A-module H with frame operator S Λ . For each f ∈ A, we have T Λ,C,C T * ,CC = K This that is a CC -controlled K-g-frame in Hilbert A-module H.
Theorem 2.6. Let Λ be a g-frame in Hilbert A-module H with frame operator S Λ . Also assume that Λ is a CC -controlled g-Bessel sequence with frame operator S CC . Then Λ is a Parseval CCcontrolled K-g-frame in Hilbert A-module H if and only if C = (S −p Λ ) * Φ and C = (S −q Λ )Ψ where Φ, Ψ are two operators in Hilbert A-module H such that Φ * Ψ = KK * and p + q = 1 where p, q ∈ R.
Proof. Assume that Λ is a Parseval CC -controlled K-g-frame in Hilbert A-module H, Hence S CC = C * S Λ C and S CC = KK * . Therefore, for each p, q ∈ R such that p + q = 1, we obtain We define Φ = (S p Λ ) * C and Ψ = (S q Λ ) * C So Conversely, let Φ and Ψ be tow operators in Hilbert A-module H such that Φ * Ψ = KK * . Suppose that C = (S −p Λ ) * Φ and C = (S −q Λ ) * Ψ are tow operators on Hilbert A-module H wherep, q ∈ R and p + q = 1, Since So, for each f ∈ H, Thus Λ is Parseval CC -controlled k − g− frame on Hilbert A-module H.

Duals of Controlled K-g-Frames
In this section, by the concept of K-g-dual pair, we present a bounded operator called dual operator and propose some known equalities and inequalities between dual operator CC -controlled K-g-frame in Hilbert A-module H.
and Λ is a CC -controlled K − g− Bessel sequence. In this cas, (Λ, Λ) is called a CC -controlled K − g− dual pair. The following results presents equivalent conditions of the CC -K-g-dual.
Proposition 3.1. Let Λ be a CC − K − g− dual for Λ. Then the following conditions are equivalent : Proof. We have It follows that Therefore, Λ is a CC -controlled K * − g− frame in Hilbert A-module H.
Theorem 3.2. Assume that C OP and D OP are the optimal bounds of Λ, respectively. Then for which A op and B op are the optimal bounds of Λ, respectively. Assume (Λ, Λ) is called a CCcontrolled K − g− dual pair and J ⊂ I. We define and we call it a dual operator.
It is clear that S J ∈ End * A (H) and S J + S J c = K where J c is the complement of J . If B 1 and B 2 are the bounds of Λ and Λ respectively, then, we have So S J is bounded. Now, by that operator S J we extend some well known equalities and inequalities for controlled K-g-frames in the following theorems.
Proof. Let f ∈ H. We can write Now, for each f ∈ H, we obtain Theorem 3.5.
Let Λ be a Parseval CC -controlled K-g-frame in Hilbert A-module H if J ⊆ I, then for each Proof. using the the proof of Theorem 3.4, we have Therefore Thus Now, for each f ∈ H, we obtain where {c i } i∈I is a sequence of positive numbers such that η := i∈I c 2 i < ∞ and 0 ≤ λ 1 , λ 2 ≤ 1. Then is a CC -controlled k − g− frame on Hilbert A-module H with bounds: Proof. For each f ∈ H, we have

Hence
(1 − λ 2 ) (C * * Since Λ is a CC -controlled K-g-frame, so Now, for the lower bound we get

Therefore
(1 + λ 2 ) (C * * (1 + λ 2 ) . Since, Proposition 4.1. Let Λ be a CC -controlled k − g− frame on Hilbert A-module H with bounds A CC and B CC . Assume that := { i ∈ End * A (H, H i ) i∈I } is a sequence of operators such that for each f ∈ H and i ∈ I, where {c i } i∈I is a sequence of positive numbers such that η := i∈I c 2 i < ∞. Then is a CCcontrolled k − g− frame on Hilbert A-module H with bounds : Proof. For each f ∈ H, we have On the other hand

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.