Int. J. Anal. Appl. (2022), 20:14 Some Properties of Controlled K-g-Frames in Hilbert C∗-Modules Rachid Echarghaoui1, M’hamed Ghiati1,∗, Mohammed Mouniane1, Mohamed Rossafi2 1Laboratory Analysis, Geometry and Applications Department of Mathematics, Faculty Of Sciences, University of Ibn Tofail, Kenitra, Morocco 2LaSMA Laboratory Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco ∗Corresponding author: mhamed.ghiati@uit.ac.ma Abstract. This paper is devoted to studying the controlled K − g−frames in Hilbert C∗−modules, some useful results are presented. Also, the concept of controlled K−g−dual frames is given. Finally, we discuss the stability problem for controlled K−g−frames in Hilbert C∗−modules. 1. 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. 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) Received: Jan. 24, 2022. 2010 Mathematics Subject Classification. 42C15. Key words and phrases. frame; g−frame; K−g−frame; controlled K−g−frames; Hilbert C∗−modules. https://doi.org/10.28924/2291-8639-20-2022-14 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-14 2 Int. J. Anal. Appl. (2022), 20:14 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 1.1. [10] 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 in product 〈., .〉A:H ×H → A such that is sesquilinear, 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 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 〈Tx,y〉A = 〈x,T∗y〉A for all y ∈K and x ∈H. Lemma 1.1. [18] Suppose that H1 and H2 two Hilbert A-Modules H and L1 ∈ End∗A(H1,H), L2 ∈ End∗A(H2,H). Then the following assertions are equivalent: (i) R(L1) ⊆R(L2), (ii) L1L∗1 ≤ λ 2L2L ∗ 2 for some λ > 0, (iii) There exists a mapping U ∈ End∗A(H1,H2) such that L1 = L2U. Moreover, if above conditions are valid, then there exists a unique operator U such that (i) ‖U‖2= inf{α > 0 L1L∗1 ≤ αL2L ∗ 2}, (ii) ker(L1) = ker(U), (iii) R(U) ⊆R(L∗2). If an operator U has a closed range, then there exists a right-inverse operator U†, (pseudo-inverse of U) in the following sense. Lemma 1.2. [17] Let U ∈ End∗A(H1,H2) be a bounded operator with closed range R(U). Then there exists a bounded operator U† ∈ End∗A(H2,H1) for which UU†x = x, x ∈R(U). Lemma 1.3. [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, (ii) There exist k > 0 such that 〈Tx,Tx〉A ≤ k〈x,x〉A for all x ∈H. Int. J. Anal. Appl. (2022), 20:14 3 Definition 1.2. [7] A family Λ := {Λi ∈ End∗A(H,Hi)}i∈I is called a g-frame in Hilbert A module H with respect to {Hi}i∈I if there exist constants 0 < A ≤ B < +∞ such that for each f ∈H, A〈f , f 〉A ≤ ∑ i∈I 〈Λif , Λif 〉A ≤ B〈f , f 〉A. Theorem 1.1. [16] Let Λ := {Λi ∈ End∗A(H,Hi)}i∈I be a g-frame in Hilbert A module H with respect to H{i∈ I} if and only if there exist constants A,B > 0 A‖〈f , f 〉A‖≤ ∥∥∥∥∥∑ i∈I 〈Λif , Λif 〉A ∥∥∥∥∥ ≤ B‖〈f , f 〉A‖ . (1.1) 2. Some Properties of Controlled K-g-Frames Now, we define controlled K-g-Frames in Hilbert C∗-modules. Definition 2.1. Let C,C′ ∈GL+(H) and K ∈ End∗A(H), we say that Λ := {Λi ∈ End ∗ A(H,Hi)}i∈I is a (C,C′)-controlled K-g-frame in Hilbert A-module H if there exist constants 0 < Acc′ < Bcc′ < +∞ such that for each f ∈H, Acc′〈K∗f ,K∗f 〉A ≤ ∑ i∈I 〈ΛiC′f , ΛiCf 〉A ≤ Bcc′〈f , f 〉A. (2.1) If the right hand of (2.1) holds, Λ is called a (C,C′)-controlled K − g−Bessel sequence in Hilbert A-module H with bound Bc. We call Λ a Parseval C,C′-controlled K-g-frame if 〈K∗f ,K∗f 〉A = ∑ i∈I 〈ΛiC′f , ΛiCf 〉A. If K = IH, then Λ is C,C′-controlled g−frame. 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 ΛiC ′ is positive for all i ∈ I, then for each f ∈H, Acc′〈f , f 〉A ≤‖(C∗Λ∗i ΛiC ′) 1 2 f‖2≤ Bcc′〈f , f 〉A. Now, let R := {(C∗Λ∗i ΛiC ′) 1 2 f : f ∈H}i∈I ⊂ ( ∑ i∈I ⊕H)`2. 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 TCC′ : R→H, TCC′((C ∗Λ∗i ΛiC ′) 1 2 f )i∈I = ∑ i∈I (C∗Λ∗i ΛiC ′f ), and T∗CC′ : H→R, 4 Int. J. Anal. Appl. (2022), 20:14 T∗CC′(f ) = ((C ∗Λ∗i ΛiC ′) 1 2 f )i∈I. Thus, the CC′-controlled g-frame operator is given by SCC′f = TCC′T ∗ CC′f = ∑ i∈I (C∗Λ∗i ΛiC ′f ). SCC′ is positive, bounded, invertible and self-adjoint. Moreover 〈SCC′f , f 〉 = ∑ i∈I 〈ΛiC′f , ΛiCf 〉 and ACC′IH < SCC′ < BCC′IH. Lemma 2.1. Let C,C′ ∈ GL+(H). A sequence Λ is a CC′-controlled g-Bessel sequence in Hilbert A-module with bound BCC′ if and only if the operator TCC′ : R→H, TCC′((C ∗Λ∗i ΛiC ′) 1 2 f )i∈I = ∑ i∈I (C∗Λ∗i ΛiC ′f ) is well defined and bounded with ‖TCC′‖≤ √ BCC′. Proof. We only need to prove the sufficient condition. Let TCC′ be a well-defined and bounded operator with ‖TCC′‖≤ √ BCC′. For each f ∈ H, we have∑ i∈I 〈ΛiC′f , ΛiCf 〉A = ∑ i∈I 〈C∗Λ∗i ΛiC ′f , f 〉A = 〈 ∑ i∈I C∗Λ∗i ΛiC ′f , f 〉A = 〈TCC′((C∗Λ∗i ΛiC ′) 1 2 f ))i∈I, f 〉A. Hence, ‖〈TCC′((C∗Λ∗i ΛiC ′) 1 2 f ))i∈I, f 〉A‖≤‖TCC′((C∗Λ∗i ΛiC ′) 1 2 f ))i∈I‖‖f‖ ≤‖TCC′‖‖((C∗Λ∗i ΛiC ′) 1 2 f ))i∈I‖‖f‖. But ‖((C∗Λ∗i ΛiC ′) 1 2 f )‖2= ∑ i∈I 〈ΛiC′f , ΛiCf 〉A, ‖((C∗Λ∗i ΛiC ′) 1 2 f )‖≤‖TCC′‖‖f‖, ‖((C∗Λ∗i ΛiC ′) 1 2 f )‖2≤‖TCC′‖2‖f‖2. It follows that ∑ i∈I 〈ΛiC′f , ΛiCf 〉A ≤ Bcc′‖〈f , f 〉A‖. Int. J. Anal. Appl. (2022), 20:14 5 and this means that Λ is a CC′-controlled g-Bessel sequence. � Lemma 2.2. Let C,C′ ∈ GL+(H). A sequence Λ is a CC′-controlled g-frame sequence in Hilbert A-module if and only if the operator TCC′ : R→H, TCC′((C ∗Λ∗i ΛiC ′) 1 2 f ) = ∑ i∈I C∗Λ∗i ΛiC ′f is well defined, bounded and surjective. Proof. Suppose that Λ is a CC′-controlled g-frame in Hilbert A-module. Since, SCC′ is surjective operator, so TCC′. For the opposite implication, by Lemma 2.1; TCC′ is a well-defined and bounded operator. So Λ is a CC′-controlled g-Bessel sequence. Now, for each f ∈ H, we have f = TCC′T † CC′f . Hence ‖f‖4 = ‖〈f , f 〉‖2 = ‖〈TCC′T † CC′f , f 〉‖ 2 = ‖〈T† CC′f ,T ∗ CC′f 〉‖ 2 ≤‖〈T† CC′f ,T † CC′f 〉‖ 2‖〈T∗CC′f ,T ∗ CC′f 〉‖ 2 ≤‖T† CC′f‖ 2‖‖T∗CC′f‖ 2 ≤‖T† CC′‖ 2‖f‖2 ∑ i∈I 〈ΛiC′f , ΛiCf 〉A. We conclude that (‖T† CC′‖ 2)−1‖〈f , f 〉‖≤ ∑ i∈I 〈ΛiC′f , ΛiCf 〉A. � Proposition 2.1. Let Λ be a CC′-controlled K-g-frames in Hilbert A-module H and K has a dense range. Suppose that (C∗Λ∗i ΛiC ′) is positive and also Vi = (C∗Λ∗i ΛiC ′) 1 2 for each i ∈ I. Then ( ⋂ i∈I ker Vi) ⊥ = H. Proof. Assume that Acc′ and Bcc′ are the frame bounds of Λ. Hence, Acc′〈K∗f ,K∗f 〉A ≤‖(C∗Λ∗i ΛiC ′) 1 2‖2≤ Bcc′〈f , f 〉A. (2.2) Since ker K∗ = (R(K))⊥ and K has a dense range, K∗ injective. Then from (2.2), for each i ∈ I, we get ⋂ i∈I ker Vi ⊆ ker K∗ = {0}. 6 Int. J. Anal. Appl. (2022), 20:14 Remark 2.1. Suppose that Λ is a CC′-controlled K-g-frame in Hilbert A with lower bound Acc′. Then, we have SCC′ > Acc ′KK∗, so by Lemma 1.1, there exists an operator U ∈ End∗A(H,R) such that TCC′U = K. (2.3) Now, we can obtain optimal frame bounds of Λ by the operator U. Indeed, it is obvious that Bop = ‖SCC′‖= ‖TCC′‖2. By Lemma 1.1, the equation (2.3) has a unique solution as U0 such that ‖U0‖2 = inf{α > 0/KK∗ ≤ αTCC′T∗CC′} = inf{α > 0/〈KK∗f , f 〉≤ α〈TCC′T∗CC′f , f 〉, f ∈H} = inf{α > 0/〈K∗f ,K∗f 〉≤ α〈T∗CC′f ,T ∗ CC′f 〉, f ∈H} = inf{α > 0/ ‖〈K∗f ,K∗f 〉‖≤ α‖〈T∗CC′f ,T ∗ CC′f 〉‖, f ∈H} = inf{α > 0/‖K∗f‖2≤ α‖T∗CC′f‖ 2, f ∈H}. Now, we have Aop = sup{A > 0 \A‖K∗f‖2≤‖T∗CC′f‖ 2, f ∈H} = (inf{α > 0 \‖K∗f‖2≤ α‖T∗CC′f‖ 2, f ∈H})−1 = U−20 . � 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 {ΛiU}i∈I, {ΛiU∗}i∈I can also CC′-controlled K-g- frame. Next, by putting connections between the operators SΛ,K,C and C′, we reach to necessary and sufficient conditions that {Λi}i∈I can be a Parseval CC′-controlled K-g-frames. Theorem 2.1. Let Λ be a CC′-controlled K-g- frame in Hilbert A module H. and U ∈ End∗A(H) such that R(U) ⊂R(K). Then Λ is a CC′-controlled U-g-frame in Hilbert A-module H. Proof. Suppose that ACC′ is a lower frame bound of Λ. Using Lemma 1.1, there exists α > 0 such that UU∗ ≤ α2KK∗. Now, for each f ∈H. We have 〈UU∗f , f 〉A ≤ α2〈KK∗f , f 〉A. We have Acc′ (α2) 〈U∗f ,U∗f 〉A ≤ Acc′〈K∗f ,K∗f 〉A ≤ ∑ i∈I 〈ΛiC′f , ΛiCf 〉A ≤ Bcc′〈f , f 〉A. � Int. J. Anal. Appl. (2022), 20:14 7 Theorem 2.2. Let Λ be a CC′-controlled K-g- frame in Hilbert A- module H. 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 {ΛiU∗}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), ‖U∗f‖> δ‖K∗f‖. Proof. Suppose that {ΛiU∗}i∈I is a CC′-controlled K-g-frame in Hilbert A module H with a lower frame bound ECC′ > 0. If BCC′ is an upper frame bound of Λ then for each f ∈R(U), we have Ecc′〈K∗f ,K∗f 〉A ≤ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A = ∑ i∈I 〈ΛiC′U∗f , ΛiCU∗f 〉A, thus Ecc′〈K∗f ,K∗f 〉A ≤ ∑ i∈I 〈ΛiC′U∗f , ΛiCU∗f 〉A ≤ Bcc′〈U∗f ,U∗f 〉A, Therefore Ecc′‖〈K∗f ,K∗f 〉A‖≤‖ ∑ i∈I 〈ΛiC′U∗f , ΛiCU∗f 〉A‖≤ Bcc′‖〈U∗f ,U∗f 〉A‖ thus Ecc′‖K∗f‖2≤ Bcc′‖U∗f‖2. so √ Ecc′ Bcc′‖K ∗f‖≤ ‖U∗f‖, for the opposite implication, for each f ∈ H, we have ‖U∗f‖= ‖(K†)∗K∗U∗f‖≤‖(K†)‖‖K∗U∗f‖. Therefore, if ACC′ is a lower frame bound of Λ, we have ACC′δ 2‖K†‖−2〈K∗f ,K∗f 〉≤ ACC′‖K†‖−2〈U∗f ,U∗f 〉 ≤ ACC′‖K∗U∗f‖2 ≤ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A. For the upper bound, it is clear that∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A ≤ Bcc′〈U∗f ,U∗f 〉A ≤ Bcc′‖U‖2〈f , f 〉A. So, (ΛiU∗)i∈I is a CC′-controlled K-g-frame in Hilbert A-module H with frame bounds ACC′δ2‖K†‖−2 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 {ΛiU∗}i∈I and {ΛiU}i∈I are CC′-controlled K-g- frame in Hilbert A- module H, then U is invertible. Proof. Suppose that {ΛiU∗}i∈I is a CC′-controlled K-g-frame in Hilbert A module H with a lower frame bound A1, and B1. Then for each f ∈H, A1〈K∗f ,K∗f 〉A ≤ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A ≤ B1〈f , f 〉A. 8 Int. J. Anal. Appl. (2022), 20:14 We have ‖A1〈K∗f ,K∗f 〉A‖≤‖ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A‖≤‖B1〈f , f 〉A‖, (2.4) hence, A1‖K∗f‖2≤‖ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉‖≤ B1‖f‖2. Since K has a dense range, K∗ is injective. Moreover, R(U) = (ker U∗)⊥ = H so U is surjective. Suppose that {ΛiU∗}i∈I is a (CC′)-controlled K-g-frame in Hilbert A module H with a lower frame bound A2 and B2. Then, for each f ∈H, A2〈K∗f ,K∗f 〉A ≤ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A ≤ B2〈f , f 〉A ‖A2〈K∗f ,K∗f 〉A‖≤‖ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A‖≤‖B2〈f , f 〉A‖ A2‖K∗f‖2≤‖ ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A‖≤ B2‖f‖2. 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∗ = IdH) such that UK = KU and U∗ commutes with C and C′. Then {ΛiU∗}i∈I is a CC′-controlled K-g-frame in Hilbert A-module H. Proof. Suppose Λ be a CC′-controlled K-g- frame in Hilbert A-module H with a lower frame bound ACC′. and BCC′ for each f ∈H, we have∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A = ∑ i∈I 〈ΛiC′U∗f , ΛiCU∗f 〉A ≤ BCC′〈U∗f ,U∗f 〉A hence, ∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A ≤ BCC′‖U∗‖2〈f , f 〉A. So, {ΛiU∗}i∈I is a CC′-controlled g-Bessel sequence. For the lower bound, we can write∑ i∈I 〈ΛiU∗C′f , ΛiU∗Cf 〉A = ∑ i∈I 〈ΛiC′U∗f , ΛiCU∗f 〉A > ACC′〈K∗U∗f ,K∗U∗f 〉A = ACC′〈(UK)∗f , (UK)∗f 〉A = ACC′〈(KU)∗f , (KU)∗f 〉A = ACC′〈U∗K∗f ,U∗K∗f 〉A = ACC′〈UU∗K∗f ,U∗K∗f 〉A = ACC′〈K∗f ,K∗f 〉A. � Int. J. Anal. Appl. (2022), 20:14 9 Theorem 2.5. Let Λ := {Λi ∈ End∗A(H,Hi)}i∈I and � := {�i ∈ End ∗ A(H,Hi)}i∈I be tow CC ′- controlled K − g− Bessel sequences in Hilbert A- module H with bounds BΛ and B� respectively. Suppose that TΛ,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. ‖K∗f‖4 = ‖〈K∗f ,K∗f 〉A‖2 = ‖〈T�,CC′T∗Λ,C,C′f ,K ∗f 〉A‖2 ≤‖T∗Λ,C,C′f‖ 2‖T∗�,CC′K ∗f‖2 = ∑ i∈I 〈ΛiC′f , ΛiCf 〉A ∑ i∈I 〈�iC′K∗f ,�iC′K∗f 〉A ≤ ∑ i∈I 〈ΛiC′f , ΛiCf 〉AB�‖〈K∗f ,K∗f 〉A‖. So, ‖〈K∗f ,K∗f 〉A‖≤ ∑ i∈I 〈ΛiC′f , ΛiCf 〉AB� � Thus B−1� ‖〈K ∗f ,K∗f 〉A‖≤ ∑ i∈I 〈ΛiC′f , ΛiCf 〉A. This that Λ is a CC′-controlled K-g-frame in Hilbert A-module H with frame operator SΛ. For each f ∈A, we have TΛ,C,C′T∗�,CC′ = K ‖Kf‖4 = ‖〈Kf,Kf 〉A‖2 = ‖〈TΛ,C,C′T∗�,CC′f ,Kf 〉A‖ 2 ≤‖T∗Λ,C,C′Kf‖ 2‖T∗�,CC′f‖ 2 = ∑ i∈I 〈ΛiC′Kf, ΛiCKf 〉A ∑ i∈I 〈�iC′f ,�iC′f 〉A ≤ ∑ i∈I 〈�iC′f ,�iCf 〉ABΛ‖〈Kf,Kf 〉A‖. Thus B−1Λ ‖〈Kf,Kf 〉A‖≤ ∑ i∈I 〈�iC′f ,�iCf 〉A. 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 SΛ. Also assume that Λ is a CC′-controlled g- Bessel sequence with frame operator SCC′. Then Λ is a Parseval CC ′- controlled 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. 10 Int. J. Anal. Appl. (2022), 20:14 Proof. Assume that Λ is a Parseval CC′-controlled K-g-frame in Hilbert A-module H, ∑ i∈I 〈ΛiC′f , ΛiCf 〉A = 〈K∗f ,K∗f 〉A = ∑ i∈I 〈f ,C∗Λ∗i ΛiC ′f 〉A = 〈f , ∑ i∈I C∗Λ∗i ΛiC ′f 〉A = 〈f ,SCC′f 〉A = 〈f ,KK∗f 〉A SCC′(f ) = ∑ i∈I C∗Λ∗i ΛiC ′(f ) = C∗( ∑ i∈I Λ∗i ΛiC ′)(f ) = C∗SΛC ′(f ). Hence SCC′ = C ∗SΛC ′ and SCC′ = KK ∗. Therefore, for each p,q ∈R such that p +q = 1, we obtain KK∗ = C∗S p Λ S q Λ C′. We define Φ = (Sp Λ )∗C and Ψ = (Sq Λ )∗C′ So Φ∗Ψ = C∗S p Λ S q Λ C′ = KK∗. 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 KK∗ = Φ∗Ψ = C∗S p Λ S q Λ C′ = C∗SΛC ′ = SCC′. So, for each f ∈H, 〈KK∗f , f 〉A = 〈K∗f ,K∗f 〉A = 〈 ∑ i∈I C∗Λ∗i ΛiC ′f , f 〉A. Thus Λ is Parseval CC′-controlled k −g− frame on Hilbert A- module H. � 3. 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. Int. J. Anal. Appl. (2022), 20:14 11 Definition 3.1. Suppose that Λ is CC′-controlled k-g-frame on Hilbert A- module H with synthesis operator TΛ,C,C′ Then Λ̃ := {Λ̃i ∈ End∗A(H,Hi)}i∈I is called a CC ′-controlled k −g− dual frame ( or brevityCC′ −Kg− dual ) for Λ if TΛ,C,C′T ∗ Λ̃,C,C′ = K, (3.1) and Λ̃ is a CC′-controlled K − g− Bessel sequence. In this cas, (Λ, Λ̃) 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 Λ̃ be a CC′−K−g− dual for Λ. Then the following conditions are equivalent : (i) TΛ,C,C′T ∗ Λ̃,C,C′ = K (ii) T Λ̃,C,C′ T∗Λ,C,C′ = K ∗ (iii) for each f , f ′ ∈H, we have 〈Kf ; f ′〉 = 〈T∗ Λ̃,C,C′ f ,T∗ Λ̃,C,C′ f 〉. Theorem 3.1. If Λ̃ be a CC′−K−g− dual for Λ, then Λ̃ is a CC′-controlled K∗−g− frame in Hilbert A- module H. Proof. We have ‖Kf‖4 = ‖〈Kf,Kf 〉A‖2 = ‖〈TΛ,C,C′T∗Λ̃,C,C′f ,Kf 〉‖ 2 = ‖〈T∗ Λ̃,C,C′ f ,T∗Λ,C,C′〉‖ 2 ≤‖T∗ Λ̃,C,C′ f‖2‖T∗Λ,C,C′Kf‖ 2 ≤ ( ∑ i∈I 〈Λ̃iC′f , Λ̃iCf 〉A)( ∑ i∈I 〈ΛiC′Kf, ΛiCKf 〉A) ≤ BC‖Kf‖2( ∑ i∈I 〈Λ̃iC′f , Λ̃iCf 〉A), It follows that B−1 C ACC′‖〈Kf,Kf 〉A‖≤ ∑ i∈I 〈Λ̃iC′f , Λ̃iCf 〉A ≤ BCC′‖〈f , f 〉A‖. Therefore, Λ̃ is a CC′-controlled K∗ −g− frame in Hilbert A- module H. � Theorem 3.2. Assume that COP and DOP are the optimal bounds of Λ̃, respectively. Then COP > B −1 op , Dop > A −1 op , for which Aop and Bop are the optimal bounds of Λ, respectively. Assume (Λ, Λ̃) is called a CC′- controlled K −g− dual pair and J ⊂I. We define SJ f := ∑ i∈J (C∗Λ∗i ΛiC ′) 1 2 (C∗Λ̃∗ i Λ̃iC ′) 1 2 f , f ∈H, 12 Int. J. Anal. Appl. (2022), 20:14 and we call it a dual operator. It is clear that SJ ∈ End∗A(H) and SJ + SJ c = K where J c is the complement of J . If B1and B2 are the bounds of Λ and Λ̃ respectively, then, we have ‖SJ f‖2 = ( sup ‖g‖=1 ‖〈SJ f ,g‖〉)2 ≤ ( sup ‖g‖=1 ( ∑ i∈J ‖〈(C∗Λ∗i ΛiC ′) 1 2 (C∗Λ̃∗ i Λ̃iC ′) 1 2 f 〉‖)2 ≤ ( ∑ i∈I ‖(C∗Λ∗i ΛiC ′) 1 2 f‖2)(( sup ‖g‖=1 ( ∑ i∈J ‖C∗Λ̃∗ i Λ̃iC ′) 1 2‖)2 ≤ B1B2‖f‖2. So SJ is bounded. Now, by that operator SJ we extend some well known equalities and inequalities for controlled K-g- frames in the following theorems. Theorem 3.3. If f ∈ H then ( ∑ i∈J〈(C ∗Λ̃∗ i Λ̃iC ′) 1 2 f , (C∗Λ∗i ΛiC ′) 1 2 Kf 〉 − ‖SJ f‖2= ( ∑ i∈J c 〈C∗Λ̃ ∗ i Λ̃iC ′)1/2f , (C∗Λ∗ i ΛiC ′)1/2Kf 〉−‖SJ cf‖2. Proof. Let f ∈H. We can write ( ∑ i∈J 〈(C∗Λ̃∗ i Λ̃iC ′) 1 2 f , (C∗Λ∗i ΛiC ′) 1 2 Kf 〉−‖SJ f‖2 = 〈K∗SJ f , f 〉−‖SJ f‖2 = 〈K∗SJ f , f 〉−〈S∗JSJ f , f 〉 = 〈(K −SJ )∗SJ f , f 〉 = 〈S∗J c (K −SJ ), f 〉 = 〈S∗J cKf,f 〉−〈S ∗ J cSJ cf , f 〉 = 〈Kf,SJ cf 〉−〈SJ cf ,SJ cf 〉 = 〈SJ cf ,Kf 〉−‖SJ cf‖2 = ( ∑ i∈J c 〈(C∗Λ̃∗ i Λ̃iC ′) 1 2 f , (C∗Λ∗ i ΛiC ′) 1 2 Kf 〉 −‖SJ cf‖2. � Theorem 3.4. Let Λ be a Parseval CC′-controlled K-g-frame in Hilbert A-module H if J ⊆ I and E ⊆ Jc, then for each f ∈H, ‖ ∑ i∈J∪E (C∗Λ∗i ΛiC ′)f‖2−‖ ∑ i∈Jc\E (C∗Λ∗i ΛiC ′)f‖2 = ‖ ∑ i∈J (C∗Λ∗i ΛiC ′)f‖2−‖ ∑ i∈Jc (C∗Λ∗i ΛiC ′)f‖2+2Re( ∑ i∈E 〈ΛiC′f , ΛiC∗KK∗f 〉). Int. J. Anal. Appl. (2022), 20:14 13 Proof. Let SΛ,Jf = ∑ i∈J (C∗Λ∗i ΛiC ′)f , therefore, SΛ,I + SΛ,Ic = KK∗. Hence S2Λ,J −S 2 Λ,Jc = S 2 Λ,J − (KK ∗ −SΛ,J)2 = KK∗SΛ,J + SΛ,JKK ∗ − (KK∗)2 = KK∗SΛ,J −SΛ,JcKK∗. Now, for each f ∈ H, we obtain ‖S2Λ,J‖ 2−‖S2Λ,Jc‖ 2= 〈KK∗SΛ,Jf , f 〉−〈SΛ,JcKK∗f , f 〉, consequently, for J ∪E instead of J: ‖ ∑ i∈J∪E (C∗Λ∗i ΛiC ′)f‖2−‖ ∑ i∈Jc\E (C∗Λ∗i ΛiC ′)f‖2 = ( ∑ i∈J∪E 〈ΛiC′f , ΛiC∗KK∗f 〉) − ∑ i∈Jc\E 〈ΛiC′f , ΛiC∗KK∗f 〉 = ( ∑ i∈J 〈ΛiC′f , ΛiC∗KK∗f 〉) − ∑ i∈Jc 〈ΛiC′f , ΛiC∗KK∗f 〉 + 2Re( ∑ i∈E 〈ΛiC′f , ΛiC∗KK∗f 〉) = ∑ i∈J (C∗Λ∗i ΛiC ′)f‖2−‖ ∑ i∈Jc (C∗Λ∗i ΛiC ′)f‖2+2Re( ∑ i∈E 〈ΛiC′f , ΛiC∗KK∗f 〉). � Theorem 3.5. Let Λ be a Parseval CC′-controlled K-g-frame in Hilbert A- module H if J ⊆ I, then for each f ∈H, ‖ ∑ i∈J (C∗Λ∗i ΛiC ′)f‖2+Re (∑ i∈Jc 〈ΛiC′f , ΛiC∗KK∗f 〉 ) = ‖ ∑ i∈Jc (C∗Λ∗i ΛiC ′)f‖2+Re (∑ i∈J 〈ΛiC′f , ΛiC∗KK∗f 〉 ) > 3 4 ‖KK∗f‖2. Proof. using the the proof of Theorem 3.4, we have S2Λ,J −S 2 Λ,Jc = KK ∗SΛ,J −SΛ,JcKK∗. Therefore S2Λ,J + S 2 Λ,Jc = 2 ( KK∗ 2 −SΛ,J )2 + (KK∗)2 2 > (KK∗)2 2 . 14 Int. J. Anal. Appl. (2022), 20:14 Thus KK∗SΛ,J + S 2 Λ,Jc + (KK ∗SΛ,J + S 2 Λ,Jc ) ∗ = KK∗SΛ,J + S 2 Λ,Jc + SΛ,JKK ∗ + S2Λ,Jc = KK∗(SΛ,J + SΛ,Jc ) + S 2 Λ,J + S 2 Λ,Jc > 3 4 (KK∗)2. Now, for each f ∈ H, we obtain ‖ ∑ i∈J (C∗Λ∗i ΛiC ′)f‖2+Re( ∑ i∈J 〈ΛiC′f , ΛiC∗KK∗f 〉) = 〈KK∗SΛ,Jf , f 〉 + 〈S2Λ,Jcf , f 〉 + 〈KK ∗ + S2Λ,Jcf , f 〉 + 〈f ,S 2 Λ,Jcf 〉> 3 4 (KK∗)2. � 4. The stability problem of controlled K −g−frames Stability of the wavelet and Gabor frames under perturbation is one of the important problems in frame theory. At first this problem was studied by Paley and Wienes for bases and then extended to frames.But the most important results are obtained by Casazza and Christensen. Here we study the perturbation of CC′-controlled K-g-frames.in Hilbert A-module H. Theorem 4.1. Let Λ be a CC′-controlled K-g- frame on Hilbert A- module H with bounds ACC′ and ACC′. Assume that � := {�i ∈ End∗A(H,Hi)i∈I} is a sequence of operators such that for each f ∈ H and i ∈ I, ‖(C∗Λ∗i ΛiC ′ −C∗ �∗i �iC ′)1/2f‖ ≤ λ1‖(C∗Λ∗i ΛiC ′)1/2f‖+λ2‖C∗ �∗i �iC ′)1/2f‖+ci〈K∗f ,K∗f 〉 1 2 where {ci}i∈I is a sequence of positive numbers such that η := ∑ i∈I c 2 i < ∞ and 0 ≤ λ1, λ2 ≤ 1. Then � is a CC′-controlled k −g− frame on Hilbert A-module H with bounds:( (1 −λ1) √ ACC′ −η 1 + λ2 )2 , ( (1 + λ1) √ BCC′ + η‖K‖ 1 −λ2 )2 . Proof. For each f ∈ H, we have ‖C∗ �∗i �iC ′)1/2f‖= ‖(C∗ �∗i �iC ′ −C∗Λ∗i ΛiC ′)1/2f + (C∗Λ∗i ΛiC ′)1/2f‖ ≤‖(C∗ �∗i �iC ′ −C∗Λ∗i ΛiC ′)1/2f‖+(C∗Λ∗i ΛiC ′)1/2f‖ ≤ λ1‖(C∗Λ∗i ΛiC ′)1/2f‖+λ2‖C∗ �∗i �iC ′)1/2f‖+ci〈K∗f ,K∗f 〉 1 2 + ‖+(C∗Λ∗i ΛiC ′)1/2f‖. Hence (1 −λ2)‖(C∗ �∗i �iC ′)1/2f‖≤ (1 + λ1)‖(C∗Λ∗i ΛiC ′)1/2f‖+ci〈K∗f ,K∗f 〉 1 2 Int. J. Anal. Appl. (2022), 20:14 15 Since Λ is a CC′-controlled K-g-frame, so ‖T∗CC′‖ 2 = ‖(C∗Λ∗i ΛiC ′)1/2f‖2 = ∑ i∈I 〈ΛiC′f , ΛiCf 〉A ≤ Bcc′〈f , f 〉A. Therefore ‖(C∗ �∗i �iC ′)1/2f‖≤ (1 + λ1)‖(C∗Λ∗i ΛiC ′)1/2f‖+ci〈K∗f ,K∗f 〉 1 2 1 −λ2 , ‖((C∗ �∗i �iC ′)1/2f‖2≤ ( (1 + λ1) √ Bcc′ + η‖K‖ 1 −λ2 ))2〈f , f 〉A. Now, for the lower bound we get ‖(C∗ �∗i �iC ′)1/2‖ = ‖C∗Λ∗i ΛiC ′)1/2f − (C∗Λ∗i ΛiC ′ −C∗ �∗i �iC ′)1/2f‖ > ‖C∗Λ∗i ΛiC ′)1/2f‖−‖(C∗Λ∗i ΛiC ′ −C∗ �∗i �iC ′)1/2f‖ > ‖C∗Λ∗i ΛiC ′)1/2f‖−λ1‖(C∗Λ∗i ΛiC ′)1/2f‖ −λ2‖C∗ �∗i �iC ′)1/2f‖−ci〈K∗f ,K∗f 〉 1 2 . Therefore (1 + λ2)‖(C∗ �∗i �iC ′)1/2f‖> (1 −λ1)‖(C∗Λ∗i ΛiC ′)1/2f‖−ci〈K∗f ,K∗f 〉 1 2 or ‖(C∗ �∗i �iC ′)1/2f‖> (1 −λ1)‖(C∗Λ∗i ΛiC ′)1/2f‖−ci〈K∗f ,K∗f 〉 1 2 (1 + λ2) . Since, ‖T∗CC′‖ 2= ‖(C∗Λ∗i ΛiC ′)1/2f‖2= ∑ i∈I 〈ΛiC′f , ΛiCf 〉A > Acc′〈K∗f ,K∗f 〉A. Thus ‖(C∗Λ∗i ΛiC ′)1/2f‖2> ( (1 −λ1) √ Acc′ −η (1 + λ2) )2〈K∗f ,K∗f 〉A. � Proposition 4.1. Let Λ be a CC′-controlled k−g− frame on Hilbert A- module H with bounds ACC′ and BCC′. Assume that � := {�i ∈ End∗A(H,Hi)i∈I} is a sequence of operators such that for each f ∈ H and i ∈ I, ‖(C∗Λ∗i ΛiC ′ −C∗ �∗i �iC ′)1/2f‖≤ ci〈K∗f ,K∗f 〉 1 2 . where {ci}i∈I is a sequence of positive numbers such that η := ∑ i∈I c 2 i < ∞. Then � is a CC ′- controlled k −g− frame on Hilbert A- module H with bounds : ( √ ACC′ −η)2, ( √ BCC′ + η‖K‖)2. 16 Int. J. Anal. Appl. (2022), 20:14 Proof. For each f ∈ H, we have ‖(C∗ �∗i �iC ′)1/2f‖ = ‖C∗Λ∗i ΛiC ′)1/2f − (C∗Λ∗i ΛiC ′ −C∗ �∗i �iC ′)1/2f‖ > ‖C∗Λ∗i ΛiC ′)1/2f‖−‖(C∗Λ∗i ΛiC ′ −C∗ �∗i �iC ′)1/2f‖ > √ ACC′〈K∗f ,K∗f 〉 1 2 −η〈K∗f ,K∗f 〉 1 2 > ( √ ACC′ −η)〈K∗f ,K∗f 〉 1 2 . Thus ‖(C∗ �∗i �iC ′)1/2f‖2> ( √ ACC′ −η)2〈K∗f ,K∗f 〉A. On the other hand ‖C∗ �∗i �iC ′)1/2f‖ = ‖(C∗ �∗i �iC ′ −C∗Λ∗i ΛiC ′)1/2f + (C∗Λ∗i ΛiC ′)1/2f‖ ≤‖(C∗ �∗i �iC ′ −C∗Λ∗i ΛiC ′)1/2f‖+‖(C∗Λ∗i ΛiC ′)1/2f‖ ≤ √ BCC′〈f , f 〉 1 2 + η〈K∗f ,K∗f 〉 1 2 A ≤ ( √ BCC′ + η‖K‖)〈f , f 〉 1 2 A. Thus ‖(C∗ �∗i �iC ′)1/2f‖2≤ ( √ BCC′ + η‖K‖)2〈f , f 〉A. � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] P. Balazs, J.-P. Antoine, A. Gryboś, Weighted and Controlled Frames: Mutual Relationship and First Numer- ical Properties, Int. J. Wavelets Multiresolut Inf. Process. 08 (2010), 109–132. https://doi.org/10.1142/ S0219691310003377. [2] R.J. Duffin, A.C. 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