SOME RESULTS ON CONTROLLED K − FRAMES IN HILBERT SPACES

Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Also K-frames have been introduced to study atomic systems with respect to bounded linear operator. In this paper, the notion of controlled K-frames will be studied and it will be shown that controlled K-frames are equivalent to K-frames under some mild conditions. Finally, the stability of controlled K-Bessel sequences under perturbation will be discussed with some examples.


Introduction
Frames in Hilbert spaces were first proposed by Duffin and Schaeffer to deal with nonharmonic Fourier series in 1952 [9] and widely studied from 1986 by Daubechies et al. [10].Now, frames play an important role not only in the theoretics also in many kinds of applications and have been widely applied in signal processing [13], sampling [11,12], coding and communications [19], filter bank theory [3], system modeling [8] and so on.
Over the years, various extentions of the frame theory have been investigated and proposed, such as the fusion frames [5,6] to deal with hierarchical data processing, g-frames [20] by Sun to deal with all existing frames as united object, oblique dual frames [11] by Elder to deal with sampling reconstructions, and etc.
The notion of K-frames were recently introduced by L. Gǎvruta to study the atomic systems with respect to a bounded linear operator K in Hilbert spaces.K-frames are more general than ordinary frames in sense that the lower frame bound only holds for the elements in the range of the K, where K is a bounded linear operator in a separable Hilbert Space H.
Recent addition to these generalized frames are the controlled frames [1].Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces, however, they were used earlier just as a tool for spherical wavelets [2].The main advantage of these frames lies in the fact that they retain all the advantages of standard frames but additionally they give a generalized way to check the frame condition while offering a numerical advantage in the sense of preconditioning.Recent developments in this direction can be found in [14][15][16][17][18] and the references therein.
In this paper, the concept of controlled K-frame will be defined and it will be shown that any controlled K-frame is equivalent to a K-frame.Finally, we will discuss the stability of controlled K-Bessel sequences under perturbation.
Throughout this paper, H is a separable Hilbert space, B(H) is the family of all bounded linear operators on H and K ∈ B(H).GL(H) denotes the set of all bounded linear operators which have bounded inverses and GL + (H) denotes the set of all positive operators in GL(H).

The paper is organized as follows:
Section 2 contains some preliminary result.In section 3, we define the concept of controlled K-frame and we will show that controlled K-frames are equivalent to K-frames.In section 4, we discuss the stability of a more general perturbation for controlled K-Bessel sequence.In section 5, we will examine with some examples the perturbation of controlled K-Bessel sequences.

Preliminaries and notations
In this section, some necessary definitions and theorems are presented.
Every non-negative operator is clearly self-adjoint.
If A ∈ B(H) is non-negative, then there exists a unique non-negative operator B such that B 2 = A.
Furthermore, B commutes with every operator that commutes with A. This will be denoted by B = A Let B + (H) be the set of positive operators on H.For self-adjoint operators T 1 and T 2 , the notation The following result is needed in the sequel, but straightforward to prove: Proposition 2.1.[1] Let T : H → H be a linear operator.Then the following conditions are equivalent: a.There exist m > 0 and M < ∞, such that mI ≤ T ≤ M I, b.T is positive and there exist m > 0 and M < ∞, such that m f 2 ≤ T There exists a self-adjoint operator A ∈ GL(H), such that f.There exist constants m > 0 and M < ∞ and operator It is well-known that all bounded operators U on a Hilbert space H are not invertible: an operator U needs to be injective and surjective in order to be invertible.For doing this, one can use right-inverse operator.
The following lemma shows that if an operator U has closed range, there exists a right-inverse operator U † in the following sense: Lemma 2.1.[7] Let H 1 and H 2 be Hilbert spaces and suppose that U : H 2 → H 1 is a bounded operator with closed range R U .Then there exists a bounded operator The operator U † in the Lemma 2.3 is called the pseudo-inverse of U .
In the literature, one will often see the pseudo-inverse of an operator U with closed range defined as the unique operator U † satisfying that If only the right inequality of the above inequality holds,{f n } n∈I is called a Bessel sequence.
Remark 2.1.The frame operator Sf = i∈I f, f i f i associated with a frame {f i } i∈I is a bounded, invertible and positive operator on H.This provides the reconstruction formulas The controlled frame operator S is defined by We call A and B the lower and upper frame bounds for K-frame, respectively.
If only the right inequality of the above inequality holds, Remark 2.2.If K = I, then K-frame are just the ordinary frame.
Remark 2.3.In the following, we will assume that R(K) is closed, since this can assume that the pseudoinverse K † of K exists.
Because of the higher generality of K-frames, some properties of ordinary frames can not hold for Kframes, such as the frame operator of a K-frame is not an isomorphism.For more differences between K-frames and ordinary frames, we refer to [21].
H is called an atomic system for K, if the following conditions are satisfied: Obviously it is a Bessel sequence, so we can define the following operator it follows that we call T, T * and S the synthesis operator, analysis operator and frame operator for K-frame {f n } ∞ n=1 , respectively.
The following theorem gives a characterization of K-frames in Hilbert spaces.
where S is the frame operator for {f n } ∞ n=1 .
Proof.The sequence {f n } ∞ n=1 is a K-frame for H with frame bounds A, B and frame operator S if and only if that is so the conclusion holds.
Remark 2.4.Frame operator of a K-frames is not invertible on H in general, but we can show that it is invertible on the subspace R(K) ⊂ H.In fact, since R(K) is closed, there exists a pseudo-inverse So, from the definition of K-frame we have which implies that S : R(K) → S(R(K)) is a homeomorphism.Furthermore, we have

Controlled K-frames
Controlled frames for spherical wavelets were introduced in [2] to get a numerically more efficient approximation algorithm and the related theory.For general frames, it was developed in [1].For getting a numerical solution of a linear system of equations Ax = b, we can solve the system of equations P Ax = P b, where P is a suitable preconditioning matrix.It was the main motivation for introducing controlled frames in [2].
Controlled frames extended to g-frames in [17] and for fusion frames in [15].In this section, the concept of controlled frames and controlled Bessel sequences will be extended to K-frames and it will be shown that controlled K-frames are equivalent K-frames.
n=1 is a K-Bessel sequence and there exist constants A > 0 and B < ∞ such that is well defined and there exists constant The following lemma characterizes C-controlled K-frames in term of their operators.
The following proposition shows that for evaluation a family {f n } ∞ n=1 ⊂ H to be a controlled K-frame it is sufficient to check just a simple operator inequality.
The following proposition shows that any controlled K-frame is a K-frame.
n=1 is a controlled K-frame with bounds A and B. Then for any f ∈ H, On the other hand for every f ∈ H, The following proposition show that any K-frame is a controlled K-frame under some conditions.
Proof.Suppose that {f n } ∞ n=1 be a K-frame with bounds A and B .Then for all f ∈ H, Hence, Therefore {f n } ∞ n=1 is a C-controlled K-frame with bounds A and B C .

Perturbation for Controlled K-Bessel Sequences
One of the most important problems in the studying of frames and its applications specially on wavelet and Gabor systems is the invariance of these systems under perturbation.At the first, the problem of perturbation studied by Paley and Wiener for bases and then extended to frames.There are many versions of perturbation of frames in Hilbert spaces, Banach space, Hilbert C * -modules and etc.In the last decade, several authors have generalized the Paley-Wiener perturbation theorem to the perturbation of frames in Hilbert spaces.The most general result of these was the following obtained by Casazza and Christensen [4].
In this section, we mainly give an important on stability of perturbation for K-frames.To do this, we have to introduce tree lemmas below first.

and only if the operator
).Moreover, if T 1 is invertible on X, then T 2 is also invertible on X, and we have Theorem 4.1.
[4] Let {x j } j∈J be a frame for a Hilbert space H with frame bounds C and D. Assume that {y j } j∈J is a sequence of H and that there exist Suppose one of the following conditions holds for any finite scalar sequence {c j } and every x ∈ H. Then {y j } j∈J is also a frame for H; Moreover, if {x j } j∈J is a Riesz basis for H and {y j } j∈J satisfies (2), then {y j } j∈J is also a Riesz basis for H.
The perturbation theorem investigated by X. Xiao, Y. Zhu, L. Gǎvruta to K-frames [21]: for any c i (i ∈ N), then {g n } ∞ n=1 is a P Q(R(K)) K-Frame for H, with frame bounds where Motivating the above theorems, we prove perturbation for controlled K-Bessel sequences.
Theorem 4.3.Suppose that {f n } +∞ n=1 ⊂ H is a C-controlled K-frame for H, with frame bounds A, B, and α, β, γ ∈ [0, ∞), such that for any c i , then {g n } +∞ n=1 is a controlled K-Bessel sequence for H with bound ( , where T, U are the synthesis operator for {f n } +∞ n=1 and {g n } +∞ n=1 , respectively.
Proof.Let {f n } +∞ n=1 be a frame for H, so by lemma 4.1, the frame operator T is bounded and T ≤ √ B.
The condition (4.1) implies that for all finite sequences {c k }, This calculation actually holds for all {c k } +∞ k=1 ∈ l 2 (N).To see this, at the first we have to prove that n=1 is a Cauchy sequence in H and therefore convergent, thus the pre-frame operator U is well defined on l 2 (N); it follows that for all In terms of the operator T, U , (4.3) states that Via lemma 4.1, this estimation shows that {g k } ∞ k=1 is a Bessel sequence with bound

Examples
In this section, we give some examples that examines the stability of the controlled K-Bessel sequences under perturbation.

Conclusion
In this article, controlled K-frames is first defined.Then, we examined the conditions that controlled K-frames are equivalent to K-frames (under certain conditions).At the end, the stability of the controlled K-Bessel sequences were checked under perturbation.
The constants A and B are called C-controlled K-frame bounds.If C = I, the C-controlled K-frame {f n } ∞ n=1 is a K-frame for H with bounds A and B.If the second part of the above inequality holds, it called C-controlled K-Bessel sequence with bound B.Definition 3.2.Let C ∈ GL + (H).A sequence {f n } ∞ n=1 ∈ H is a C-controlledBessel sequence for H if and only if the operator