Perturbation and Stability of Continuous Operator Frames in Hilbert C ∗-Modules

$e concept of frames in Hilbert spaces is a new theory which was introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. $is theory was reintroduced and developed by Daubechieset al. [2]. In 1993, Aliet al. [3] introduced the concept of continuous frames in Hilbert spaces. Gabardo and Han in [4] called these kinds of frames, frames associated with measurable spaces. In 2000, Frank and Larson [5] introduced the notion of frames in Hilbert C∗-modules as a generalization of frames in Hilbert spaces. $e theory of continuous frames has been generalized in Hilbert C∗-modules. For more details, see [6–25]. $e aim of this paper is to extend the results of Rossafi and Akhlidj [23], given for Hilbert C∗-module in a discrete case. In the following, we briefly recall the definitions and basic properties of C∗-algebra and Hilbert A-modules. Our references for C∗-algebras are [26, 27]. For C∗-algebra A, if a ∈ A is positive, we write a≥ 0, and A denotes the set of positive elements of A.


Introduction and Preliminaries
e concept of frames in Hilbert spaces is a new theory which was introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series.
is theory was reintroduced and developed by Daubechieset al. [2].
In 1993, Aliet al. [3] introduced the concept of continuous frames in Hilbert spaces. Gabardo and Han in [4] called these kinds of frames, frames associated with measurable spaces.
In the following, we briefly recall the definitions and basic properties of C * -algebra and Hilbert A-modules. Our references for C * -algebras are [26,27]. For C * -algebra A, if a ∈ A is positive, we write a ≥ 0, and A + denotes the set of positive elements of A.
Definition 1 (see [26]). Let A be unital C * -algebra and H be 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 〈., .〉 A : H × H ⟶ A, such that it is sesquilinear and positive definite and respects the module action. In the other words, (i) 〈x, x〉 A ≥ 0, for all x ∈ H, and 〈x, x〉 A � 0 if and only if x � 0. (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) A , for all x ∈ H.

Characterisation of Continuous Operator
Frame for End * A (H) Let X be a Banach space, (Ω, μ) a measure space, and f: Ω ⟶ X be a measurable function. Integral of Banachvalued function f has been defined by Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions [30,31]. Since every C * -algebra and Hilbert C * -module are Banach spaces, we can use this integral and its properties. Let (Ω, μ) be a measure space, U and V be two Hilbert C * -modules over a unital C * -algebra and V w w∈Ω is a family of submodules of V. End * A (U, V w ) is the collection of all adjointable A-linear maps from U into V w .
We define the following: For any x � x w w∈Ω and y � y w w∈Ω , the A-valued inner product is defined by 〈x, y〉 � Ω 〈x w , y w 〉 A dμ(w) and the norm is defined by ‖x‖ � ‖〈x, x〉‖ (1/2) . In this case, l 2 (Ω, V w ω∈Ω ) is an Hilbert C * -module [32].
there is a pair of constants 0 < ], δ such that for any x ∈ H, e constants ] and δ are called continuous operator frame bounds.
If ] � δ, we call this continuous operator frame a continuous tight operator frame, and if ] � δ � 1, it is called a continuous Parseval operator frame.
If only the right-hand inequality of (4) is satisfied, we call Λ � Λ w w∈Ω the continuous Bessel operator frame for End * A (H) with Bessel bound δ. e continuous frame operator S of Λ on H is defined by e continuous frame operator S is a bounded, positive, self-adjoint, and invertible.  Proof. We just prove the case that T w + R w w∈Ω is a continuous operator frame for End * A (H). On the one hand, for each x ∈ H, we have

Perturbation and Stability of Continuous
en, From (8) and (9), we get erefore, T w + R w w∈Ω is a continuous operator frame for End * A (H).
□ Theorem 3. Let T w w∈Ω be a continuous operator frame for End * A (H) with bounds ] and δ and let R w w∈Ω ⊂ End * A (H). e following statements are equivalent: Proof. Suppose that R w w∈Ω is a continuous operator frame for End * A (H) with bound η and ρ. en for all x ∈ H, we have In the same way, we have For (12), we take ξ � min From (12), we have en, Also, we have From (12), we have en, So, From (18) and (22), we give that then n k�1 α k T k,w w∈Ω is a continuous operator frame for End * A (H) and conversely.
Proof. For every x ∈ H, we have

Journal of Mathematics
Hence Hence, Also, we have en, en for λ � (]/δ p ), we have which ends the proof.
If there exists a constant λ > 0 such that for each x ∈ H and k � 1, . . . , n, we have 6

Journal of Mathematics
Since, for any x ∈ H, we have Hence is give that n k�1 R k,w w∈Ω is a continuous operator frame for End * A (H).

Characterisation of Continuous K-Operator Frames for End * A (H)
Definition 3. Let K ∈ End * A (H). A family of adjointable operators T w w∈Ω on a Hilbert A-module H is said to be a continuous K-operator frame for End * A (H), if there exists two positive constants ], δ > 0 such that e numbers ] and δ are called, respectively, lower and upper bound of the continuous K-operator frame. e continuous K-operator frame is called a ]-thight if: If ] � 1, it is called a normalised tight continuous K-operator frame or a Parseval continuous K-operator frame. e continuous K-operator frame is standard if for every x ∈ H, the sum (40) converges in norm.

Remark 1. For any K ∈ End *
A (H), every continuous operator frame is a continuous K-operator frame.
Indeed, for any K ∈ End * A (H), we have Let T w w∈Ω be a continuous operator frame with bounds ] and δ, then Hence T w w∈Ω is a continuous K-operator frame with bounds ]‖K‖ − 2 and δ.
Let T w w∈Ω be a continuous K-operator for End * A (H). We define the operator R: H ⟶ l 2 (H), e operator R is called the analysis operator of the continuous K-operator frame T w w∈Ω , and its adjoint is defined as follows: e operators R is called the synthesis operator of the continuous K-operator frame T w w∈Ω .
By composing R and R * , we obtain the operator

Journal of Mathematics
It is easy to show that the operator S K is positive and selfadjoint.
Proof. Suppose that T w w∈Ω is a continuous K-operator frame.
From the definition of continuous K-operator frame, (47) holds.
Conversely, assume that (47) holds. e frame operator S K is positive and self-adjoint; then We have for any Using Lemma 2, there exist two constants τ, ξ > 0 such that is proves that T w w∈Ω is a continuous K-operator frame for End * A (H).

Perturbation and Stability of Continuous K-Operator Frames for End
en R w w∈Ω is a continuous K-operator frame for End * A (H).
Proof. For every x ∈ H, we have en, Hence Also, for all x ∈ H, we have Hence us So, Hence is give that R w w∈Ω is a continuous K-operator frame for End * A (H).
Proof. Let T w w∈Ω be a continuous K-operator frame with bounds ] and δ. en for any x ∈ H, we have