Int. J. Anal. Appl. (2022), 20:1 Controlled K −g−Fusion Frames in Hilbert C∗−Modules Mohamed Rossafi1,∗, Fakhr-dine Nhari2 1LaSMA Laboratory Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco 2Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty Of Sciences, University of Ibn Tofail, Kenitra, Morocco ∗Corresponding author: rossafimohamed@gmail.com Abstract. Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g−fusion frame and controlled K−g−fusion frame in Hilbert C∗−modules and we give some properties. Also, we study the perturbation problem of controlled K − g−fusion frame. Moreover, an illustrative example is presented to support the obtained results. 1. Introduction Frames for Hilbert spaces were introduced by Duffin and Schaefer [4] in 1952 to study some deep problems in nonharmonic Fourier series by abstracting the fundamental notion of Gabor [6] for signal processing. Many generalizations of the concept of frame have been defined in Hilbert C∗-modules [5,7,9,13– 17]. Controlled frames in Hilbert spaces have been introduced by P. Balazs [3] to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Rashidi and Rahimi [10] are introduced the concept of Controlled frames in Hilbert C∗−modules. Received: Nov. 15, 2021. 2010 Mathematics Subject Classification. 42C15. Key words and phrases. fusion frame; g-fusion frame; K−g−fusion frame; controlled K−g−fusion frames; Hilbert C∗−modules. https://doi.org/10.28924/2291-8639-20-2022-1 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-1 2 Int. J. Anal. Appl. (2022), 20:1 The paper is organized in the following manner. In section 3, we introduced the notion of g−fusion frames and controlled g−fusion frames in Hilbert C∗−modules and estabilish some properties. Section 4 is devoted to introduce the concept of controlled K −g−fusion frames in Hilbert C∗−modules and gives some results, finally in section 5 we study the perturbation of controlled K −g−fusion frames. 2. Preliminaires Let A be a unital C∗−algebra, let J be countable index set. Throughout this paper H and L are countably generated Hilbert A−modules and {Hj}j∈J is a sequence of submodules of L. For each j ∈ J, End∗A(H,Hj) is the collection of all adjointable A−linear maps from H to Hj, and End ∗ A(H,H) is denoted by End∗A(H). Also let GL +(H) be the set of all positive bounded linear invertible operators on H with bounded inverse. Definition 2.1. [8] 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 ×H → A, such that is sesquilinear, positive definite and respects the module action. In the other words, (i) 〈f , f 〉≥ 0 for all f ∈ H and 〈f , f 〉 = 0 if and only if f = 0. (ii) 〈af + g,h〉 = a〈f ,h〉 + 〈g,h〉 for all a ∈A and f ,g,h ∈ H. (iii) 〈f ,g〉 = 〈g,f 〉∗ for all f ,g ∈ H. For f ∈ H, we define ||f || = ||〈f , f 〉|| 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 a C∗-algebra A, we have |a| = (a∗a) 1 2 and the A-valued norm on H is defined by |f | = 〈f , f 〉 1 2 for f ∈ H. Define l2({Hj}j∈J) by l2({Hj}j∈J) = {{fj}j∈J : fj ∈ Hj, || ∑ j∈J 〈fj, fj〉|| < ∞}. With A−valued inner product is given by 〈{fj}j∈J,{gj}j∈J〉 = ∑ j∈J 〈fj,gj〉, l2({Hj}j∈J) is a Hilbert A−module. The following lemmas was used to proof our results: Lemma 2.1. [1] If φ : A → B is a ∗−homomorphism between C∗−algebras, then φ is increasing, that is, if a ≤ b, then φ(a) ≤ φ(b). Lemma 2.2. [2] Let T ∈ End∗A(H,L) and H,L are Hilberts A−modules.The following statemnts are multually equivalent: (i) T is surjective. Int. J. Anal. Appl. (2022), 20:1 3 (ii) T∗ is bounded below with respect to the norm, i.e., there is m > 0 such that ||T∗f || ≥ m||f || for all f ∈ L. (iii) T∗ is bounded below with respect to the inner product, i.e, there is m ′ > 0 such that 〈T∗f ,T∗f 〉≥ m ′ 〈f , f 〉 for all f ∈ L. Lemma 2.3. [1] Let H and L are two Hilbert A-modules and T ∈ End∗A(H,L). Then: (i) If T is injective and T has closed range, then the adjointable map T∗T is invertible and ‖(T∗T )−1‖−1 ≤ T∗T ≤‖T‖2. (ii) If T is surjective, then the adjointable map TT∗ is invertible and ‖(TT∗)−1‖−1 ≤ TT∗ ≤‖T‖2. Lemma 2.4. [2] Let H be a Hilbert A-module over a C∗-algebra A, and T ∈ End∗A(H) such that T∗ = T . The following statements are equivalent: (i) T is surjective. (ii) There are m,M > 0 such that m‖f‖≤‖Tf‖≤ M‖f‖, for all f ∈ H. (iii) There are m′,M′ > 0 such that m′〈f , f 〉≤ 〈Tf,Tf 〉≤ M′〈f , f 〉 for all f ∈ H. Lemma 2.5. [12] Let H be a Hilbert A-module. If T ∈ End∗A(H), then 〈Tf,Tf 〉≤ ‖T‖2〈f , f 〉, ∀f ∈ H. Lemma 2.6. [18] Let E,H and L be Hilbert A−modules, T ∈ End∗A(E,L) and T ′ ∈ End∗A(H,L). Then the following two statements are equivalent: (1) T ′ (T ′ )∗ ≤ λTT∗ for some λ > 0; (2) There exists µ > 0 such that ‖(T ′ )∗z‖≤ µ‖T∗z‖ for all z ∈ L. Lemma 2.7. [11] Let {Wj}j∈J be a sequence of orthogonally complemented closed submodules of H and T ∈ End∗A(H) invertible, if T ∗TWj ⊂ Wj for each j ∈ J, then {TWj}j∈J is a sequence of orthogonally complemented closed submodules and PWjT ∗ = PWjT ∗PTWj. 3. Controlled g−fusion frame in Hilbert C∗−modules Firstly we give the definition of g−fusion frame in Hilbert C∗−modules. Definition 3.1. [11] Let {Wj}j∈J be a sequence of closed submodules orthogonally complemented of H, {vj}j∈J be a family of weights in A, ie., each vj is positive invertible element frome the center of A and Λj ∈ End∗A(H,Hj) for each j ∈ J. We say that Λ = {Wj, Λj,vj}j∈J is a g−fusion frame for H if there exists 0 < A ≤ B < ∞ such that A〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉≤ B〈f , f 〉, ∀f ∈ H. (3.1) 4 Int. J. Anal. Appl. (2022), 20:1 The constants A and B are called the lower and upper bounds of the g−fusion frame, respectively. If A = B then Λ is called tight g−fusion frame and if A = B = 1 then we say Λ is a Parseval g−fusion frame. The operator S : H → H defined by Sf = ∑ j∈J v2j PWj Λ ∗ j ΛjPWjf , ∀f ∈ H. Is called g−fusion frame operator. Now we define the notion of (C,C ′ )−controlled g−fusion frame in Hilbert C∗−modules. Definition 3.2. Let C, C ′ ∈ GL+(H), {Wj}j∈J be a sequence of closed submodules orthogonally complemented of H, {vj}j∈J be a family of weights in A, i.e., each vj is a positive invertible element frome the center of A and Λj ∈ End∗A(H,Hj) for each j ∈ J. We say that ΛCC′ = {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H if there exists 0 < A ≤ B < ∞ such that A〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ B〈f , f 〉, ∀f ∈ H. (3.2) The constants A and B are called the lower and upper bounds of the (C,C ′ )−controlled g−fusion frame, respectively. When A = B, the sequence Λ CC ′ = {Wj, Λj,vj}j∈J is called (C,C ′ )−controlled tight g−fusion frame, and when A = B = 1, it is called a (C,C ′ )−controlled Parseval g−fusion frame. If only upper inequality of (3.2) hold, then Λ CC ′ is called an (C,C ′ )−controlled g−fusion bessel sequence for H. Example 3.1. Let l∞ be the set of all bounded complex-valued sequences. For any u = {uj}j∈N, v = {vj}j∈N ∈ l∞, we have uv = {ujvj}j∈N,u∗ = {uj}j∈N, ||u|| = sup j∈N |uj|. Then A = {l∞, ||.||} is a C∗−algebra. Let H = C0 be the set of all sequences converging to zero. For any u, v ∈ H we define 〈u,v〉 = uv∗ = {ujvj}j∈N. Then H is a Hilbert A−module. Now let {ej}j∈N be the standard orthonormal basis of H. We construct Hj = span{e1,e2, ...,ej} and Wj = span{ej} for each j ∈N. Define Λj : H → Hj by Λj(f ) = ∑j k=1〈f , ej√ j 〉ek. The adjoint operator Λ∗j : Hj → H define by Λ ∗ j (g) = ∑j k=1〈g, ek√ j 〉ej. And the projection orthogonal PWj define by PWj (f ) = 〈f ,ej〉ej. Int. J. Anal. Appl. (2022), 20:1 5 Let us define Cf = 2f and C ′ f = 1 2 f . Then for any f ∈ H, we have 〈ΛjPWjCf, ΛjPWjC ′ f 〉 = 〈 2 √ j 〈f ,ej〉 j∑ k=1 ek, 1 2 √ j 〈f ,ej〉 j∑ k=1 ek〉 = 1 j 〈f ,ej〉〈ej, f 〉〈 j∑ k=1 ek, j∑ k=1 ek〉 = 1 j 〈f ,ej〉〈ej, f 〉 j∑ k=1 ||ek||2 = 1 j 〈f ,ej〉〈ej, f 〉j = 〈f ,ej〉〈ej, f 〉. Therefore, for each f ∈ H,∑ j∈N 〈ΛjPWjCf, ΛjPWjC ′ f 〉 = ∑ j∈N 〈f ,ej〉〈ej, f 〉 = 〈f , f 〉. Hence {Wj, Λj, 1}j∈N is a (C,C ′ )−controlled Parseval g−fusion frame for H. Suppose that Λ CC ′ be a (C,C ′ )−controlled g−fusion bessel sequence for H. The bounded linear operator T (C,C ′ ) : l2({Hj}j∈J) → H define by T (C,C ′ ) ({fj}j∈J) = ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j fj, ∀{fj}j∈J ∈ l 2({Hj}j∈J). (3.3) is called the synthesis operator for the (C,C ′ )−controlled g−fusion frame Λ CC ′. The adjoint operator T∗ (C,C ′ ) : H → l2({Hj}j∈J) given by T∗ (C,C ′ ) (g) = {vjΛjPWj (C ′ C) 1 2 g}j∈J (3.4) is called the analysis operator for the (C,C ′ )−controlled g−fusion frame Λ CC ′. When C and C ′ commute with each other, and commute with the operator PWj Λ ∗ j ΛjPWj, for each j ∈ J, then the (C,C ′ )−controlled g−fusion frame operator S (C,C ′ ) : H → H is defined as S (C,C ′ ) (f ) = T (C,C ′ ) T∗ (C,C ′ ) (f ) = ∑ j∈J v2j C ′ PWj Λ ∗ j ΛjPWjCf, ∀f ∈ H. (3.5) And we have 〈S (C,C ′ ) (f ), f 〉 = ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉, ∀f ∈ H. (3.6) From now we assume that C and C ′ commute with each other, and commute with the operator PWj Λ ∗ j ΛjPWj, for each j ∈ J Lemma 3.1. Let Λ CC ′ be a (C,C ′ )−controlled g−fusion frame for H. Then the (C,C ′ )−controlled g−fusion frame operator S (C,C ′ ) is positive, self-adjoint and invertible. 6 Int. J. Anal. Appl. (2022), 20:1 Proof. For each f ∈ H we have S (C,C ′ ) (f ) = ∑ j∈J v2j C ′ PWj Λ ∗ j ΛjPWjCf Then ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉 = 〈 ∑ j∈J v2j C ′ PWj Λ ∗ j ΛjPWjCf,f 〉 = 〈S(C,C′)(f ), f 〉. Since Λ CC ′ is a (C,C ′ )−controlled g−fusion frame for H, then A〈f , f 〉≤ 〈S (C,C ′ ) (f ), f 〉≤ B〈f , f 〉, ∀f ∈ H (3.7) It is clear that S (C,C ′ ) is positive, bounded and linear operator. On the other hand for each f , g ∈ H 〈S (C,C ′ ) (f ),g〉 = 〈 ∑ j∈J v2j C ′ PWj Λ ∗ j ΛjPWjCf,g〉 = 〈f , ∑ j∈J v2j CPWj Λ ∗ j ΛjPWjC ′ g〉 = 〈f ,S (C ′ ,C) (g)〉. That implies S∗ (C,C ′ ) = S (C ′ ,C) . Also as C and C ′ commute with each other, and commute with the operator PWj Λ ∗ j ΛjPWj, for each j ∈ J, we have S(C,C′) = S(C′,C). So the (C,C ′ )−controlled g−fusion frame operator S (C,C ′ ) is self-adjoint. And from inequality (3.7) we have AIH ≤ S(C,C′) ≤ BIH. (3.8) Therefore, the (C,C ′ )−controlled g−fusion frame operator S (C,C ′ ) is invertible. � We estabilish an equivalent definition of (C,C ′ )−controlled g−fusion frame. Theorem 3.1. Λ CC ′ = {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H. If and only if there exists two constants 0 < A ≤ B < ∞ such that A||f ||2 ≤ || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| ≤ B||f ||2, ∀f ∈ H. (3.9) Proof. If Λ CC ′ be a (C,C ′ )−controlled g−fusion frame for H, then we have inequality (3.9). Converselly, assume that (3.9) holds. From (3.4), the (C,C ′ )−controlled g−fusion frame operator S (C,C ′ ) is positive, self-adjoint and invertible. Then we have for all f ∈ H 〈(S (C,C ′ ) ) 1 2 f , (S (C,C ′ ) ) 1 2 f 〉 = 〈S (C,C ′ ) f , f 〉 = ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉. (3.10) Using (3.9) and (3.10), we conclude that √ A||f || ≤ ||S 1 2 (C,C ′ ) f || ≤ √ B||f ||, ∀f ∈ H. So by lemma 2.4, Λ CC ′ is a (C,C ′ )−controlled g−fusion frame for H. � Int. J. Anal. Appl. (2022), 20:1 7 Theorem 3.2. Let {Wj, Λj,vj}j∈J be a g−fusion frame for H with frame operator S and let C, C ′ ∈ GL+(H). Then {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H. Proof. Let {Wj, Λj,vj}j∈J be a g−fusion frame for H with frame bounds A and B. Then for each f ∈ H A〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉≤ B〈f , f 〉 (3.11) We have || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| = ||〈S (C,C ′ ) f , f 〉|| = ||C||.||C ′ ||.||〈Sf,f 〉||, (3.12) Using (3.11) and (3.12), we conclude A||C||.||C ′ ||||〈f , f 〉|| ≤ || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| ≤ B||C||.||C ′ ||||〈f , f 〉||, ∀f ∈ H. Therefore, {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H with bounds A||C||.||C ′ || and B||C||.||C ′ ||. � Remark 3.1. When C = C ′ we say that the sequence {Wj, Λj,vj}j∈J is a C2−controlled g−fusion frame for H. Theorem 3.3. Let C ∈ GL+(H). The sequence {Wj, Λj,vj}j∈J is a g−fusion frame for H if and only if {Wj, Λj,vj}j∈J is a C2−controlled g−fusion frame for H. Proof. Suppose that {Wj, Λj,vj}j∈J is a g−fusion frame for H. with bounds A and B. Then A〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉≤ B〈f , f 〉, ∀f ∈ H. We have for each f ∈ H,∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjCf 〉≤ B〈Cf,Cf 〉≤ B||C|| 2〈f , f 〉. (3.13) On the other hand for each f ∈ H A〈f , f 〉 = A〈C−1Cf,C−1Cf 〉≤ A||C−1||2〈Cf,Cf 〉 ≤ ||C−1||2 ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjCf 〉. (3.14) So from (3.13) and (4.1), we have A||C−1||−2〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjCf 〉≤ B||C|| 2〈f , f 〉, ∀f ∈ H. 8 Int. J. Anal. Appl. (2022), 20:1 We conclude that {Wj, Λj,vj}j∈J is a C2−controlled g−fusion frame for H. Converselly, Let {Wj, Λj,vj}j∈J be a C2−controlled g−fusion frame for H with bounds A ′ and B ′ . Then for all f ∈ H, A ′ 〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjCf 〉≤ B ′ 〈f , f 〉 We have for each f ∈ H, ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉 = ∑ j∈J v2j 〈ΛjPWjCC −1f , ΛjPWjCC −1f 〉 ≤ B ′ 〈C−1f ,C−1f 〉 ≤ B ′ ||C−1||2〈f , f 〉. (3.15) Also for each f ∈ H, A ′ 〈C−1f ,C−1f 〉≤ ∑ j∈J v2j 〈ΛjPWjCC −1f , ΛjPWjCC −1f 〉 = ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉 And A ′ ||(C−1C−1)−1||−1〈f , f 〉≤ A ′ 〈C−1f ,C−1f 〉≤ ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉 (3.16) From (3.15) and (3.16), we have A ′ ||(C−2)−1||−1〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉≤ B ′ ||C−1||2〈f , f 〉, ∀f ∈ H. Hence {Wj, Λj,vj}j∈J is a g−fusion frame for H. � Theorem 3.4. Let C, C ′ ∈ GL+(H), and C, C ′ commute with each other and commute with PWj Λ ∗ j ΛjPWj for all j ∈ J. Then ΛCC′ = {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled g−fusion bessel sequence for H with bound B if and only if the operator T (C,C ′ ) : l2({Hj}j∈J) → H given by T (C,C ′ ) ({gj}j∈J) = ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j gj, ∀{gj}j∈J ∈ l 2({Hj}j∈J). is well defined and bounded operator with, ||T (C,C ′ ) || ≤ √ B. Proof. Let Λ CC ′ is a (C,C ′ )−controlled g−fusion bessel sequence with bound B for H. As a result of theorem 3.1, || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| ≤ B||f ||2, ∀f ∈ H. (3.17) Int. J. Anal. Appl. (2022), 20:1 9 For any {gj}j∈J ∈ l2({Hj}j∈J), ||T (C,C ′ ) ({gj}j∈J)|| = sup ||f ||=1 ||〈T (C,C ′ ) ({gj}j∈J), f 〉|| = sup ||f ||=1 ||〈 ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j gj, f 〉|| = sup ||f ||=1 || ∑ j∈J 〈vj(CC ′ ) 1 2 PWj Λ ∗ j gj, f 〉|| = sup ||f ||=1 || ∑ j∈J 〈gj,vjΛjPWj (CC ′ ) 1 2 f 〉|| ≤ sup ||f ||=1 || ∑ j∈J 〈gj,gj〉|| 1 2 || ∑ j∈J v2j 〈ΛjPWj (CC ′ ) 1 2 f , ΛjPWj (CC ′ ) 1 2 f 〉|| 1 2 = sup ||f ||=1 || ∑ j∈J 〈gj,gj〉|| 1 2 || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| 1 2 ≤ sup ||f ||=1 || ∑ j∈J 〈gj,gj〉|| 1 2 √ B||f || = √ B||{gj}j∈J||. Therefore, the sum ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j gj is convergent, and we have ||T (C,C ′ ) ({gj}j∈J)|| ≤ √ B||{gj}j∈J|| Hence the operator T (C,C ′ ) is well defined, bounded and ||T (C,C ′ ) || ≤ √ B. For the converse, suppose that the operator T (C,C ′ ) is well defined, bounded and ||T (C,C ′ ) || ≤ √ B. For all f ∈ H, we have || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| = || ∑ j∈J v2j 〈C ′ PWj Λ ∗ j ΛjPWjCf,f 〉|| = || ∑ j∈J v2j 〈(CC ′ ) 1 2 PWj Λ ∗ j ΛjPWj (CC ′ ) 1 2 f , f 〉|| = ||〈T (C,C ′ ) ({gj}j∈J), f 〉|| ≤ ||T (C,C ′ ) ||||{gj}j∈J||||f || = ||T (C,C ′ ) |||| ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| 1 2 ||f || Where gj = vjΛjPWj (CC ′ ) 1 2 f . Hence || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| 1 2 ≤ √ B||f || Then || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| ≤ B||f ||2 (3.18) 10 Int. J. Anal. Appl. (2022), 20:1 The adjoint operator of T (C,C ′ ) is given by T∗ (C,C ′ ) (g) = {vjΛjPWj (CC ′ ) 1 2 g}j∈J, ∀g ∈ H. And we have for each f ∈ H || ∑ j∈J 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| = || ∑ j∈J v2j 〈ΛjPWj (CC ′ ) 1 2 f , ΛjPWj (CC ′ ) 1 2 f 〉|| = ||〈T∗ (C,C ′ ) (f ),T∗ (C,C ′ ) (f )〉|| = ||T∗ (C,C ′ ) (f )||2 Frome (3.18), we have ||T∗ (C,C ′ ) (f )|| ≤ √ B||f ||, ∀f ∈ H. So, T∗ (C,C ′ ) is bounded A−linear operator, then there exist a constant M > 0 such that 〈T∗ (C,C ′ ) f ,T∗ (C,C ′ ) f 〉≤ M〈f , f 〉, ∀f ∈ H. Hence ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ M〈f , f 〉. ∀f ∈ H. This give that Λ CC ′ is a (C,C ′ )−controlled g−fusion bessel sequence for H. � Theorem 3.5. Let {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled g−fusion frame for H with bounds A and B, with operator frame S (C,C ′ ) . Let θ ∈ End∗A(H) be injective and has a closed range. Suppose that θ commute with C, C ′ and PWj for all j ∈ J. Then {Wj, Λjθ,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H. Proof. Let {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled g−fusion frame for H with bounds A and B, then A〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ B〈f , f 〉, ∀f ∈ H. For each f ∈ H, we have∑ j∈J v2j 〈ΛjθPWjCf, ΛjθPWjC ′ f 〉 = ∑ j∈J v2j 〈ΛjPWjCθf, ΛjPWjC ′ θf 〉 ≤ B〈θf,θf 〉 ≤ B||θ||2〈f , f 〉 (3.19) And A〈θf,θf 〉≤ ∑ j∈J v2j 〈ΛjθPWjCf, ΛjθPWjC ′ f 〉, By lemma 2.3, we have A||(θ∗θ)−1||−1〈f , f 〉≤ A〈θf,θf 〉 Int. J. Anal. Appl. (2022), 20:1 11 So A||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈J v2j 〈ΛjθPWjCf, ΛjθPWjC ′ f 〉 (3.20) Using (3.19) and (3.20) we conclude that A||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈J v2j 〈ΛjθPWjCf, ΛjθPWjC ′ f 〉≤ B||θ||2〈f , f 〉, ∀f ∈ H. Therefore {Wj, Λjθ,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H. � Theorem 3.6. Let {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled g−fusion frame for H. with bounds A and B. Let θ ∈ End∗A(L,H) be injective and has a closed range. Suppose that θ commute with ΛjPWjC and ΛjPWjC ′ for all j ∈ J. Then {Wj,θΛj,vj}j∈J be a (C,C ′ )−controlled g−fusion frame for H. Proof. Let {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled g−fusion frame for H with bounds A and B, then A〈f , f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ B〈f , f 〉, ∀f ∈ H. We have for each f ∈ H∑ j∈J v2j 〈θΛjPWjCf,θΛjPWjC ′ f 〉≤ ||θ||2 ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉 ≤ B||θ||2〈f , f 〉 (3.21) On the other hand, A〈θf,θf 〉≤ ∑ j∈J v2j 〈θΛjPWjCf,θΛjPWjC ′ f 〉 = ∑ j∈J v2j 〈ΛjPWjCθf, ΛjPWjC ′ θf 〉 By lemma 2.3, we have A||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈J v2j 〈θΛjPWjCf,θΛjPWjC ′ f 〉 (3.22) Using (3.21) and (3.22) , we conclude that A||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈J v2j 〈θΛjPWjCf,θΛjPWjC ′ f 〉≤ B||θ||2〈f , f 〉, ∀f ∈ H. Hence, {Wj,θΛj,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for H. � Under wich conditions a (C,C ′ )−controlled g−fusion frame for H with H a C∗−module over a unital C∗−algebras A is also a (C,C ′ )−controlled g−fusion frame for H with H a C∗−module over a unital C∗−algebras B. the following theorem answer this questions. We teak in next theorem Hj ⊂ H, ∀j ∈ J. 12 Int. J. Anal. Appl. (2022), 20:1 Theorem 3.7. Let (H,A,〈., .〉A) and (H,B,〈., .〉B) be two Hilbert C∗−modules and let φ : A → B be a ∗−homomorphisme and θ be a map on H such that 〈θf,θg〉B = φ(〈f ,g〉A) for all f , g ∈ H. Suppose that Λ CC ′ = {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled g−fusion frame for (H,A,〈., .〉A) with frame operator SA and lower and upper bounds A and B respectively. If θ is surjective such that θΛjPWj = ΛjPWjθ for each j ∈ J and θC = Cθ and θC ′ = C ′ θ. Then {Wj, Λj,φ(vj)}j∈J is a (C,C ′ )−controlled g−fusion frame for (H,B,〈., .〉B) with frame operator SB and lower and upper bounds A and B respectively and 〈SBθf,θg〉B = φ(〈SAf ,g〉A). Proof. Since θ is surjective, then for every g ∈ H there exists f ∈ H such that θf = g. Using the definition of (C,C ′ )−controlled g−fusion frame for (H,A,〈., .〉A) we have A〈f , f 〉A ≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉A ≤ B〈f , f 〉A By lemma 2.1 we have φ ( A〈f , f 〉A ) ≤ φ (∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉A ) ≤ φ ( B〈f , f 〉A ) Frome the definition of ∗−homomorphisme we have Aφ ( 〈f , f 〉A ) ≤ ∑ j∈J φ(v2j )φ ( 〈ΛjPWjCf, ΛjPWjC ′ f 〉A ) ≤ Bφ ( 〈f , f 〉A ) Using the relation betwen θ and φ we get A〈θf,θf 〉B ≤ ∑ j∈J φ(vj) 2〈θΛjPWjCf,θΛjPWjC ′ f 〉B ≤ B〈θf,θf 〉B Since θΛjPWj = ΛjPWjθ for each j ∈ J and θC = Cθ and θC ′ = C ′ θ we have A〈θf,θf 〉B ≤ ∑ j∈J φ(vj) 2〈ΛjPWjCθf, ΛjPWjC ′ θf 〉B ≤ B〈θf,θf 〉B Therefore, A〈g,g〉B ≤ ∑ j∈J φ(vj) 2〈ΛjPWjCg, ΛjPWjC ′ g〉B ≤ B〈g,g〉B, ∀g ∈ H. This implies that {Wj, Λj,φ(vj)}j∈J is a (C,C ′ )−controlled g−fusion frame for (H,B,〈., .〉B) with bounds A and B. Moreover we have φ ( 〈SAf ,g〉A ) = φ ( 〈 ∑ j∈J v2j C ′ PWj Λ ∗ j ΛjPWjCf,g〉A ) = ∑ j∈J φ(vj) 2φ ( 〈ΛjPWjCf, ΛjPWjC ′ g〉A ) Int. J. Anal. Appl. (2022), 20:1 13 = ∑ j∈J φ(vj) 2〈θΛjPWjCf,θΛjPWjC ′ g〉B = 〈 ∑ j∈J φ(vj) 2C ′ PWj Λ ∗ j ΛjPWjCθf,θg〉B = 〈SBθf,θg〉B. � 4. (C,C ′ )−controlled K −g−fusion frames in Hilbert C∗−modules Firstly we give the definition of K −g−fusion frame in Hilbert C∗−modules. Definition 4.1. [11] Let A be a unital C∗−algebra and H be a countably generated Hilbert A−module. let ( vj ) j∈J be a family of weights in A,i.e.,each vj is a positive invertible element frome the center of A, let ( Wj ) j∈J be a collection of orthogonally complemented closed submodules of H. Let ( Kj ) j∈J a sequence of closed submodules of K and Λj ∈ End∗A(H,Kj) for each j ∈ J and K ∈ End ∗ A(H). We say Λ = (Wj, Λj,vj)j∈J is K − g−fusion frame for H with respect to (Kj)j∈J if there exist real constants 0 < A ≤ B < ∞ such that A〈K∗f ,K∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjf , ΛjPWjf 〉≤ B〈f , f 〉, ∀f ∈H. (4.1) The constants A and B are called a lower and upper bounds of K −g−fusion frame, respectively. If the left-hand inequality of (4.1) is an equality, we say that Λ is a tight K−g−fusion frame. If K = IH then Λ is a g−fusion frame and if K = IH and Λj = PWj for any j ∈ J, then Λ is a fusion frame for H Definition 4.2. Let C, C ′ ∈ GL+(H) and K ∈ End∗A(H). {Wj}j∈J be a sequence of closed submodules orthogonally complemented of H, {vj}j∈J be a family of weights in A, i.e., each vj is a positive invertible element frome the center of A and Λj ∈ End∗A(H,Hj) for each j ∈ J. We say ΛCC′ = {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled K −g−fusion frame for H if there exists 0 < A ≤ B < ∞ such that A〈K∗f ,K∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ B〈f , f 〉, ∀f ∈ H. (4.2) The constants A and B are called a lower and upper bounds of (C,C ′ )−controlled K − g−fusion frame, respectively. If the left-hand inequality of (4.2) is an equality, we say that Λ CC ′ is a tight (C,C ′ )−controlled K −g−fusion frame for H. Remark 4.1. If Λ CC ′ is a (C,C ′ )−controlled K−g−fusion frame for H with bounds A and B we have AKK∗ ≤ S (C,C ′ ) ≤ BIH. (4.3) From equality (3.6) and inequality (4.3) we have 14 Int. J. Anal. Appl. (2022), 20:1 Proposition 4.1. Let K ∈ End∗A(H), and ΛCC′ be a (C,C ′ )−controlled g−fusion bessel sequence for H. Then Λ CC ′ is a (C,C ′ )−controlled K −g−fusion frame for H if and only if there exist a constant A > 0 such that AKK∗ ≤ S (C,C ′ ) where S (C,C ′ ) is the frame operator for Λ CC ′. Theorem 4.1. Let Λ CC ′ = {Wj, Λj,vj}j∈J and ΓCC′ = {Vj, Γj,uj}j∈J be two (C,C ′ )−controlled g−fusion bessel sequences for H with bounds B1 and B2, respectively. Suppose that TΛ CC ′ and TΓ CC ′ be their synthesis operators such that TΓ CC ′T ∗ Λ CC ′ = K∗ for some K ∈ End∗A(H). Then, both Λ CC ′ and Γ CC ′ are (C,C ′ )−controlled K and K∗ −g−fusion frames for H, respectively. Proof. For each f ∈ H, we have 〈K∗f ,K∗f 〉 = 〈TΓ CC ′T ∗ Λ CC ′ f ,TΓCC′ T∗Λ CC ′ f 〉≤ ||TΓCC′ || 2〈T∗Λ CC ′ ,T ∗ Λ CC ′ f 〉 ≤ B2 ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉, Hence B−12 〈K ∗f ,K∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉. This means that Λ CC ′ is a (C,C ′ )−controlled K − g−fusion frame for H. Similarly, Γ CC ′ is a (C,C ′ )−controlled K∗ −g−fusion frame for H with the lower bound B−11 . � Theorem 4.2. Let U ∈ End∗A(H) be an invertible operator on H and ΛCC′ = {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled K −g−fusion frame for H for some K ∈ End∗A(H). Suppose that U ∗UWj ⊂ Wj, ∀j ∈ J and C, C ′ commute with U. Then Γ CC ′ = {UWj, ΛjPWjU ∗,vj}j∈J is a (C,C ′ )−controlled UKU∗ −g−fusion frame for H. Proof. Since Λ CC ′ is a (C,C ′ )−controlled K −g−fusion frame for H, ∃ A,B > 0 such that A〈K∗f ,K∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ B〈f , f 〉, ∀f ∈ H. Also, U is an invertible linear operator on H, so for any j ∈ J, UWj is closed in H. Now, for each f ∈ H, using lemma 2.7, we obtain ∑ j∈J v2j 〈ΛjPWjU ∗PUWjCf, ΛjPWjU ∗PUWjC ′ f 〉 = ∑ j∈J v2j 〈ΛjPWjU ∗Cf, ΛjPWjU ∗C ′ f 〉 = ∑ j∈J v2j 〈ΛjPWjCU ∗f , ΛjPWjC ′ U∗f 〉 ≤ B〈U∗f ,U∗f 〉 ≤ B||U||2〈f , f 〉. Int. J. Anal. Appl. (2022), 20:1 15 On the other hand, for each f ∈ H A〈(UKU∗)∗f , (UKU∗)∗f 〉 = A〈UK∗U∗f ,UK∗U∗f 〉 ≤ A||U||2〈K∗U∗f ,K∗U∗f 〉 ≤ ||U||2 ∑ j∈J v2j 〈ΛjPWjC(U ∗f ), ΛjPWjC ′ (U∗f )〉 = ||U||2 ∑ j∈J v2j 〈ΛjPWjU ∗Cf, ΛjPWjU ∗C ′ f 〉 ≤ ||U||2 ∑ j∈J v2j 〈ΛjPWjU ∗PUWjCf, ΛjPWjU ∗PUWjC ′ f 〉, Then A ||U||2 〈(UKU∗)∗f , (UKU∗)∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjU ∗PUWjCf, ΛjPWjU ∗PUWjC ′ f 〉 Therefore, Γ CC ′ is a (C,C ′ )−controlled UKU∗ −g−fusion frame for H. � Theorem 4.3. Let U ∈ End∗A(H) be an invertible operator on H and ΓCC′ = {UWj, ΛjPWjU ∗,vj}j∈J be a (C,C ′ )−controlled K−g−fusion frame for H for some K ∈ End∗A(H). Suppose that U ∗UWj ⊂ Wj, ∀j ∈ J and C, C ′ commute with U. Then Λ CC ′ = {Wj, Λj,vj}j∈J is a (C,C ′ )−controlled U−1KU − g−fusion frame for H. Proof. Since Γ CC ′ = {UWj, ΛjPWjU ∗,vj}j∈J is a (C,C ′ )−controlled K −g−fusion frame for H, ∃ A, B > 0 such that A〈K∗f ,K∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjU ∗PUWjCf, ΛjPWjU ∗PUWjC ′ f 〉≤ B〈f , f 〉. ∀f ∈ H. Let f ∈ H, we have A〈(U−1KU)∗f , (U−1KU)∗f 〉 = A〈U∗K∗(U−1)∗f ,U∗K∗(U−1)∗f 〉 ≤ A||U||2〈K∗(U−1)∗f ,K∗(U−1)∗f 〉 ≤ ||U||2 ∑ j∈J v2j 〈ΛjPWjU ∗PUWjC(U −1)∗f , ΛjPWjU ∗PUWjC ′ (U−1)∗f 〉 ≤ ||U||2 ∑ j∈J v2j 〈ΛjPWjU ∗C(U−1)∗f , ΛjPWjU ∗C ′ (U−1)∗f 〉 = ||U||2 ∑ j∈J v2j 〈ΛjPWjU ∗(U−1)∗Cf, ΛjPWjU ∗(U−1)∗C ′ f 〉 = ||U||2 ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉. Then, for each f ∈ H, we have A ||U||2 〈(U−1KU)∗f , (U−1KU)∗f 〉≤ ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉. 16 Int. J. Anal. Appl. (2022), 20:1 Also, for each f ∈ H, we have ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉 = ∑ j∈J v2j 〈ΛjPWjCU ∗(U−1)∗f , ΛjPWjC ′ U∗(U−1)∗f 〉 = ∑ j∈J v2j 〈ΛjPWjU ∗C(U−1)∗f , ΛjPWjU ∗C ′ (U−1)∗f 〉 = ∑ j∈J v2j 〈ΛjPWjU ∗PUWjC(U −1)∗f , ΛjPWjU ∗PUWjC ′ (U−1)∗f 〉 ≤ B〈(U−1)∗f , (U−1)∗f 〉 ≤ B||U−1||2〈f , f 〉. Thus, Λ CC ′ is a (C,C ′ )−controlled U−1KU −g−fusion frame for H. � Theorem 4.4. Let K ∈ End∗A(H) be an invertible operator on H and ΛCC′ = {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled g−fusion frame for H with frame bounds A, B and S (C,C ′ ) be the associ- ated (C,C ′ )−controlled g−fusion frame operator. Suppose that for all j ∈ J, T∗TWj ⊂ Wj, where T = KS−1 (C,C ′ ) and C, C ′ commute with T . Then {KS−1 (C,C ′ ) Wj, ΛjPWjS −1 (C,C ′ ) K∗,vj}j∈J is a (C,C ′ )−controlled K − g−fusion frame for H with the corresponding (C,C ′ )−controlled g−fusion frame operator KS−1 (C,C ′ ) K∗. Proof. We now T = KS−1 (C,C ′ ) is invertible on H and T∗ = (KS−1 (C,C ′ ) )∗ = S−1 (C,C ′ ) K∗. For each f ∈ H, we have 〈K∗f ,K∗f 〉 = 〈S (C,C ′ ) S−1 (C,C ′ ) K∗f ,S (C,C ′ ) S−1 (C,C ′ ) K∗f 〉 ≤ ||S (C,C ′ ) ||2〈S−1 (C,C ′ ) K∗f ,S−1 (C,C ′ ) K∗f 〉 ≤ B2〈S−1 (C,C ′ ) K∗f ,S−1 (C,C ′ ) K∗f 〉. Now for each f ∈ H, we get ∑ j∈J v2j 〈ΛjPWjT ∗PTWjC(f ), ΛjPWjT ∗PTWjC ′ (f )〉 = ∑ j∈J v2j 〈ΛjPWjT ∗C(f ), ΛjPWjT ∗C ′ (f )〉 = ∑ j∈J v2j 〈ΛjPWjCT ∗(f ), ΛjPWjC ′ T∗(f )〉 ≤ B〈T∗f ,T∗f 〉 ≤ B||T ||2〈f , f 〉 ≤ B||S−1 (C,C ′ ) ||2||K||2〈f , f 〉 ≤ B A2 ||K||2〈f , f 〉. Int. J. Anal. Appl. (2022), 20:1 17 On the other hand, for each f ∈ H, we have∑ j∈J v2j 〈ΛjPWjT ∗PTWjC(f ), ΛjPWjT ∗PTWjC ′ (f )〉 = ∑ j∈J v2j 〈ΛjPWjT ∗C(f ), ΛjPWjT ∗C ′ (f )〉 = ∑ j∈J v2j 〈ΛjPWjCT ∗(f ), ΛjPWjC ′ T∗(f )〉 ≥ A〈T∗f ,T∗f 〉 = A〈S−1 (C,C ′ ) K∗f ,S−1 (C,C ′ ) K∗f 〉 ≥ A B2 〈K∗f ,K∗f 〉. Thus {KS−1 (C,C ′ ) Wj, ΛjPWjS −1 (C,C ′ ) K∗,vj}j∈J is a (C,C ′ )−controlled K −g−fusion frame for H. For each f ∈ H, we have∑ j∈J v2j C ′ PTWj (ΛjPWjT ∗)∗(ΛjPWjT ∗)PTWjCf = ∑ j∈J v2j C ′ PTWjTPWj Λ ∗ j (ΛjPWjT ∗)PTWjCf = ∑ j∈J v2j C ′ (PWjT ∗PTWj ) ∗Λ∗j Λj(PWjT ∗PTWj)Cf = ∑ j∈J v2j C ′ TPWj Λ ∗ j ΛjPWjT ∗Cf = ∑ j∈J v2j TC ′ PWj Λ ∗ j ΛjPWjCT ∗f = T ( ∑ j∈J v2j C ′ PWj Λ ∗ j ΛjPWjCT ∗f ) = TS (C,C ′ ) T∗(f ) = KS−1 (C,C ′ ) K∗(f ). This implies that KS−1 (C,C ′ ) K∗ is the associated (C,C ′ )−controlled g−fusion frame operator. � The next theorem we give an equivqlent definition of (C,C ′ )−controlled K −g−fusion frame. Theorem 4.5. Let K ∈ End∗A(H). Then ΛCC′ is a (C,C ′ )−controlled K −g−fusion frame for H if and only if there exist constants A, B > 0 such that A||K∗f ||2 ≤ || ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉|| ≤ B||f ||2, ∀f ∈ H. (4.4) Proof. Evidently, every (C,C ′ )−controlled K −g−fusion frame for H satisfies (4.4). For the converse, we suppose that (4.4) holds. For any {fj}j∈J ∈ l2({Hj}j∈J), || ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j fj|| = sup ||g||=1 ||〈 ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j fj,g〉|| = sup ||g||=1 || ∑ j∈J 〈vj(CC ′ ) 1 2 PWj Λ ∗ j fj,g〉|| 18 Int. J. Anal. Appl. (2022), 20:1 = sup ||g||=1 || ∑ j∈J 〈fj,vjΛjPWj (CC ′ ) 1 2 g〉|| ≤ sup ||g||=1 || ∑ j∈J 〈fj, fj〉|| 1 2 || ∑ j∈J v2j 〈ΛjPWj (CC ′ ) 1 2 g, ΛjPWj (CC ′ ) 1 2 g〉|| 1 2 = sup ||g||=1 || ∑ j∈J 〈fj, fj〉|| 1 2 || ∑ j∈J v2j 〈ΛjPWjCg, ΛjPWjC ′ g〉|| 1 2 ≤ sup ||g||=1 || ∑ j∈J 〈fj, fj〉|| 1 2 √ B||g|| = √ B||{fj}j∈J||. Thus the series ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j fj converges in H unconditionally. Since 〈Tf,{fj}j∈J〉 = ∑ j∈J 〈vjΛjPWj (CC ′ ) 1 2 f , fj〉 = 〈f , ∑ j∈J vj(CC ′ ) 1 2 PWj Λ ∗ j fj〉. T is adjointable. Now for each f ∈ H we have 〈Tf,Tf 〉 = ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉≤ ||T ||2〈f , f 〉. On the other hand the left-hand inequality of (4.4) gives ||K∗f ||2 ≤ 1 A ||Tf ||2, ∀f ∈ H. Then the lemma 2.6 implies that there exist a constant µ > 0 such that KK∗ ≤ µT∗T, And hence 1 µ 〈K∗f ,K∗f 〉≤ 〈Tf,Tf 〉 = ∑ j∈J v2j 〈ΛjPWjCf, ΛjPWjC ′ f 〉, ∀f ∈ H. Consequently, Λ CC ′ is a (C,C ′ )−controlled K −g−fusion frame for H. � 5. perturbation of (C,C ′ )−controlled K −g−fusion frame in Hilbert C∗−modules Theorem 5.1. Let Λ CC ′ = {Wj, Λj,vj}j∈J be a (C,C ′ )−controlled K − g−fusion frame for H with frame bounds A, B and Γj ∈ End∗A(H,Hj). Suppose that for each f ∈ H, ||((vjΛjPWj −ujΓjPVj )(CC ′ ) 1 2 f )j∈J|| ≤λ1||(vjΛjPWj (CC ′ ) 1 2 f )j∈J||+ λ2||(ujΓjPVj (CC ′ ) 1 2 f )j∈J|| + �||K∗f ||. where 0 < λ1,λ2 < 1 and � > 0 such that � < (1 −λ1) √ A. Then {Wj, Γj,uj}j∈J is a (C,C ′ )−controlled K −g−fusion frame for H. Int. J. Anal. Appl. (2022), 20:1 19 Proof. We have for each f ∈H || ∑ j∈J u2j 〈ΓjPVjCf, ΓjPVjC ′ f 〉|| 1 2 = ||(ujΓjPVj (CC ′ ) 1 2 f )j∈J|| = ||(ujΓjPVj (CC ′ ) 1 2 f )j∈J + (vjΛjPWj (CC ′ ) 1 2 f )j∈J − (vjΛjPWj (CC ′ ) 1 2 f )j∈J|| ≤ ||((ujΓjPVj −vjΛjPWj )(CC ′ ) 1 2 f )j∈J|| + ||(vjΛjPWj (CC ′ ) 1 2 f )j∈J|| ≤ (λ1 + 1)||(vjΛjPWj (CC ′ ) 1 2 f )j∈J|| + λ2||(ujΓjPVj (CC ′ ) 1 2 f )j∈J|| + �||K∗f ||. So (1 −λ2)||(ujΓjPVj (CC ′ ) 1 2 f )j∈J|| ≤ (λ1 + 1) √ B||f || + �||K∗f ||. Then ||(ujΓjPVj (CC ′ ) 1 2 f )j∈J|| ≤ (λ1 + 1) √ B||f || + �||K∗f || 1 −λ2 ≤ ( (λ1 + 1) √ B + �||K|| 1 −λ2 )||f ||. Hence || ∑ j∈J u2j 〈ΓjPVjCf, ΓjPVjC ′ f 〉|| ≤ ( (λ1 + 1) √ B + �||K|| 1 −λ2 )2||f ||2. On the other hand for each f ∈H || ∑ j∈J u2j 〈ΓjPVjCf, ΓjPVjC ′ f 〉|| 1 2 = ||(ujΓjPVj (CC ′ ) 1 2 f )j∈J|| = ||((ujΓjPVj −vjΛjPWj )(CC ′ ) 1 2 f )j∈J + (vjΛjPWj (CC ′ ) 1 2 f )j∈J|| ≥ ||(vjΛjPWj (CC ′ ) 1 2 f )j∈J|| − ||((ujΓjPVj −vjΛjPWj )(CC ′ ) 1 2 f )j∈J|| ≥ (1 −λ1)||(vjΛjPWj (CC ′ ) 1 2 f )j∈J|| −λ2||(ujΓjPVj (CC ′ ) 1 2 f )j∈J||− �||K∗f ||. Hence || ∑ j∈J u2j 〈ΓjPVjCf, ΓjPVjC ′ f 〉|| ≥ ( (1 −λ1) √ A− � 1 + λ2 )2||K∗f ||2. By theorem 4.5, we conclude that {Vj, Γj,uj}j∈J is a (C,C ′ )−controlled K−g−fusion frame for H. � 20 Int. J. Anal. Appl. (2022), 20:1 Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] A. 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Xu, On majorization and range inclusion of operators on Hilbert C∗-modules, Linear Multilinear Algebra. 66 (2018), 2493–2500. https://doi.org/10.1080/03081087.2017.1402859. https://doi.org/10.1090/S0002-9939-06-08498-X https://doi.org/10.1142/S0219691310003377 https://doi.org/10.1142/S0219691310003377 https://doi.org/10.1090/S0002-9947-1952-0047179-6 https://doi.org/10.2307/2372552 https://doi.org/10.1142/S0219691308002458 https://doi.org/10.1142/S0219691317500382 https://doi.org/10.28924/2291-8639-19-2021-836 https://doi.org/10.1090/S0002-9947-1973-0355613-0 https://doi.org/10.1090/S0002-9947-1973-0355613-0 https://doi.org/10.1142/S1793557120500606 https://doi.org/10.1142/S1793557120500606 https://doi.org/10.1080/03081087.2017.1402859 1. Introduction 2. Preliminaires 3. Controlled g-fusion frame in Hilbert C-modules 4. (C,C')-controlled K-g-fusion frames in Hilbert C-modules 5. perturbation of (C,C')-controlled K-g-fusion frame in Hilbert C-modules References