EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1237-1248 ISSN 1307-5543 – ejpam.com Published by New York Business Global Chatterjee and Extension of Chatterjee Fixed Point Theorems on Operators on Hilbert C*-Modules Rashwan. A. Rashwan1,∗, Howida Adel AlFran2, Asmaa Fangary3, Saleh Omran3 1 Department of Mathematics, Faculty of Science, Assuit University, Assuit, Egypt 2 Department of Mathematics, Al-Leith University College, Umm Al Qura University, Kingdom of Saudi Arabia 3 Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt Abstract. In this paper we consider some fixed point theorem (such as Chatterjee and extension of Chatterjee) in operators of Hilbert C∗-modules, based on a definition of valued operator Hilbert C*-modules normed space. Also We give some examples to clear our definitions. 2020 Mathematics Subject Classifications: 47H10, 46L05, 46L08 Key Words and Phrases: Fixed Point Theorems, C∗-algebra, Operators on Hilbert C∗-modules 1. Introduction Hilbert C∗-modules consider a mathematical objects where generalize the notion of a Hilbert space by allowing the inner product to take values in a (commutative, unital) C∗-algebra rather than in the field of complex numbers. Hilbert C∗-modules were first introduced in 1953 by Kaplansky [5]. Later, the theory was developed independently by Paschke [12] and Rieffel [16] where the research on Hilbert C∗-modules began in the 70,s in the work of the induced representations of C∗-algebras by M. A. Rieffel [16] also Kas- parov [6] introduced the definition of KK-theory by using Hilbert C∗-modules C*-algebra is a main subject in the functional analysis and the operator theory which play fundamental role in noncommutative geometry and theoretical physics, especially the quantum mechanics Ma and et al. [20], introduced the concept of C∗-algebra-valued metric spaces. The main idea consists in using the set of all positive elements of a unital C∗-algebra instead ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4071 Email addresses: rashwan10@gmail.com (R. A. Rashwan ), hafran@uqu.edu.sa (H. Adel AlFran), Asmaa.fangary44@yahoo.com (A. Fangary),salehomran@yahoo.com (S. Omran) http://www.ejpam.com 1237 © 2021 EJPAM All rights reserved. R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1238 of the set of real numbers. They presented some fixed point results for mapping under contractive or expansive conditions in these spaces. Later, Ma and et al. [21], introduced the concept of C∗-algebra-valued b-metric spaces and proved some fixed point theorems such as Banach and Kannan type fixed point theorems.For other results on C∗-algebra- valued b-metric spaces and C∗-algebra-valued-metric spaces, see [4, 13, 15, 18, 22]. An element x ∈ A is a positive element, denote it by x ⪰ 0 , if x ∈ Ah and σ(x) ⊂ [0,+∞], where σ(x) is the spectrum of x and Ah = {x ∈ A : x∗ = x}. Using positive elements, one can define a partial ordering ⪯ on Ah as follows: x ⪯ y if and only if y−x ⪰ 0. From now on, by A+ we denote the set {x ∈ A : x ⪰ 0} and |x| = (x∗x) 1 2 . 2. Preliminaries In this section, we begin with some basic notations and definition C∗-algebra and fixed point theory that will be very important and useful in the sequal. Definition 1. [9] A Banach ∗-algebra is a ∗-algebra A together with a complete submul- tiplicative norm such that ∥ab∥ ≤ ∥a∥∥b∥ (for all a, b ∈ A). A C∗algebra is a Banach ∗-algebra such that ∥a∗a∥ = ∥a∥2 (for all a ∈ A) . Definition 2. [9] An element a ∈ A is positive element, if a = a∗ and σ(a) ⊆ R+ , where σ(a) is the spectrum of a, we denote A+ the set of all positive element in A. Definition 3. [8, 19] A pre-Hilbert C∗-module E over a C∗-algebra A , is a right A-module together with an A-valued inner product < ., . >: E × E −→ A satisfying the conditions: (1) < x, x >⪰ 0 for all x ∈ E; (2) < x, x >= 0 if and only if x = 0; (3) < x,αy + βz >= α < x, y > +β < x, z > for all x, y, z ∈ E , α, β ∈ C; (4) < x, ya >=< x, y > a for all x, y ∈ E , a ∈ A; (5) < x, y >∗=< y, x > for all x, y ∈ E. Definition 4. [8] The norm of an element e ∈ E is defined as ∥x∥E := √ ∥ < x, x > ∥R, where ∥.∥R is the R-valued norm. If a pre-Hilbert A -module is complete with respect to its norm,it is said to be a Hilbert A -module. Example 1. Every C∗-algebra A is a Hilbert A-module over itself when equipped with the A-valued inner product given simply by < a, b >= a∗b, (a, b ∈ A). Definition 5. [19] Let E be a Hilbert A-module. A map T : E −→ E is said to be adjointable if there exists a map T ∗ : E −→ E satisfying R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1239 < x, Ty >=< T ∗x, y > for all x, y ∈ E. Definition 6. [3] An element T ∈ l(E) is positive if for every x ∈ E we have < Tx, x >A⪰ 0 and we write it by T ⪰ 0 and we denote the set l(E)+ = {T ∈ E ; T ⪰ 0}, we define a partial ordering relation on l(E)+ as if T1, T2 ∈ l(E), T1 ⪯l(E) T2 if and only if T2 − T1 ∈ l(E)+ Definition 7. [3] l(E) = {T : E −→ E} is the set of all adjiontable linear operators with ∥T∥ = sup{∥Tx∥E ; ∥x∥E ≤ 1} is a C∗-algebra. 3. Main Results Definition 8. Let l(E)+ be a subset of l(E). l(E)+ is called Cone of l(E) if and only if : (1) l(E)+ ∩ (−l(E)+) = {0l(E)}, (0l(E) is the zero vector); (2) l(E)+ is closed in l(E); (3) Ta+ Sb ∈ l(E)+ ; aT + bS ∈ l(E)+ a, b ∈ A , Tλ+ Sβ ∈ l(E)+ : λ, β ∈ C ; (4) l(E)+ · l(E)+ ⊆ l(E)+ . Definition 9. An l(E)-valued metric on a set X is a function dl(E) : X×X −→ l(E) such that for all x, y and z in X the following conditions are hold: (1) dl(E)(x, y) ⪰ 0; (2) dl(E)(x, y) = 0 if and only if x = y; (3) dl(E)(x, y) = dl(E)(y, x); (4) dl(E)(x, y) ⪯ dl(E)(x, z) + dl(E)(z, y). Then the triple (X, l(E), dl(E)) is called an l(E)-valued metric space. Definition 10. [20] Let X be a nonempty set. Suppose the mapping d : X × X −→ A satisfies: (1) 0A ⪯ d(x, y) for all x, y ∈ X and d(x, y) = 0A if and only if x = y. (2) d(x, y) = d(y, x) for all x, y ∈ X . (3) d(x, y) ⪯ d(x, z) + d(z, y) for all x, y, z ∈ X. Then d is called a C∗-algebra-valued metric on X and (X,A, d) is a C∗-algebra-valued metric space. R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1240 Definition 11. Let (X, l(E), dl(E)) be an l(E)- valued metric spacs. Suppose that xn ⊂ X and x ∈ X If for any εl(E) ≻ 0l(E) (where 0l(E) is the zero element in l(E) ) there exists N ∈ N such that for all n > N , dl(E)(xn, x) ⪯ εl(E), then {xn} is said to be converge with respect to l(E), and {xn} converges to x and x is the limit of {xn}. We denote it by limn−→+∞{xn} = x . If for any εl(E) ≻ 0l(E) there exists N ∈ N such that for all n,m > N , d(xn, xm) ⪯ εl(E), then {xn} is said to be a Cauchy with respect to l(E). We say (X, l(E), dl(E)) is a complete l(E)- valued metric spacs if every Cauchy sequence with respect to l(E) is convergent. Lemma 1. A sequence xn ⊂ X is convergence if ∥xn∥ −→ 0 forall n > N such that N ∈ N. Example 2. Let X = A⊕n, E = A⊕n and l(E) = {T : A⊕n −→ A⊕n : T (a1, a2, ..., an) = (Ta1, Ta2, ..., Tan)}. Define d((a1, a2, ..., an), (b1, b2, ..., bn)) = (∥Ta1 − Tb1∥R, ∥Ta2 − Tb2∥R, ..., ∥Tan − Tbn∥R)IA, where (a1, a2, ..., an), (b1, b2, ..., bn) ∈ A⊕n and IA is the identity element of A . It is easy to verify that dl(E) is an l(E) -valued metric space and (X,A⊕n, dl(E)) is a complete l(E) -valued metric space, since A is complete. Definition 12. let (X, l(E)) is an l(E)-metric space, we define the open ball on X Bl(E)(a, ϵl(E)) = {x ∈ X; ∥x− a∥ ≺ ϵl(E)} Definition 13. Suppose that (X, dl(E)) is l(E)-metric space, let x ∈ X then a neighhbor- hood of x is any set containing Bl(E)(x, ϵl(E)) for some ϵl(E) ≻ 0l(E). Definition 14. Suppose that (X, dl(E)) is l(E)-metric space, a subset U ⊂ X is open if for every x ∈ U there exist an open ball Bl(E)(a, ϵl(E)) such that x ∈ Bl(E)(x, ϵl(E)) ⊂ U . Motivaied by the idea in [7],[17],[9], we give the following definations. Definition 15. Let X be vector space, if the function ∥.∥l(E) : X −→ l(E) has the following properties: (1) ∥x∥l(E) ⪰ 0 i.e ∥x∥l(E) is a positive operator, ∥x∥l(E) = 0 if and only if x = 0; (2) ∥λx∥l(E) = |λ|∥x∥l(E) ; λ ∈ C; (3) ∥x+ y∥l(E) ⪯ ∥x∥l(E) + ∥y∥l(E). Then ∥.∥ is said to be l(E)-valued norm defined on X, and (X, ∥.∥) is said to be l(E)-valued normed l(E) space. Also we will set the relation between l(E)-valued metric space and l(E)-valued normed space as follow dl(E)(x, y) = ∥x− y∥l(E). R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1241 Definition 16. Let X be a vector space over a field (F = C,R) we say that X is a right l(E)-vector space if satisfy: (1) (x+ y)T = xT + yT ; (3) x(T1 + T2) = xT1 + xT2; (3) (xS)T = x(ST ). Where x, y ∈ X and S, T ∈ l(E). Lemma 3.2 Let X be a right l(E)-vector space then, ∥xT∥l(E) ⪯ ∥x∥∥T∥l(E). Proof. ∥xT∥2 = sup∥x∥=1{< xT, xT >, x ∈ E} ≤ ∥x∥∥T∥l(E). Definition 17. Let A be C∗-algebra,and l(E) be an l(E)-normed spac. We say that l(E) is right A-module if the mapping is right module multiplication (a, T ) 7−→ xa of A× l(E) −→ l(E) such that the following axioms are satisfied: (1) For each fixed a ∈ A the map (a, T ) −→ Ta is linear on l(E): T ∈ l(E) ; (2) For each fixed T ∈ l(E) the map (a, T ) −→ Ta is linear on A; (3) For all a1, a2 ∈ A and all T ∈ l(E) we have that (Ta1)a2 = T (a1a2). Example 3. If we define the norm ∥x∥l(E) = ∥x∥Il(E) (where Il(E) is the identity operator of l(E) ) then we have that l(E) with this norm is l(E)-norm. Lemma 2. If T is positive if and only if T ∗ is positive. Proof. Let ∗ : A −→ A is ∗-homomorphism. if T ∗ is positive implies < T ∗x, x >⪰ 0 implies < x, Tx >⪰ 0 implies < x, Tx >∗⪰ 0 implies < Tx, x >⪰ 0 implies T is positive. ⇐= if T is positive implies < Tx, x >⪰ 0 implies < x, T ∗x >⪰ 0 implies < x, T ∗x >∗⪰ 0 implies < T ∗x, x >⪰ 0 implies T ∗ ⪰ 0 implies T ∗ is positive. Lemma 3. If S is positive operator then for any operator T implies T ∗ST is positive operator. Proof. Since S ⪰ 0, we can write S = R∗R, for any R ∈ (lE) implies T ∗(R∗R)T = (T ∗R∗)(RT ) = (RT )∗(RT ) ⪰ 0 Definition 18. A sequence {xn} in X is said to be convergent if for every ϵ > 0, there is a natural number N such that for n > N we have R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1242 ∥xn − x∥ ⪯l(E) ϵIl(E) (where Il(E) the identity operator of l(E) ). Definition 19. A sequence {xn} in X is said to be a Cuachy sequence if for every ϵ > 0, there is a natural number N such that for n,m > N we have ∥xn − xm∥ ⪯l(E) ϵIl(E). Lemma 4. A sequence {xn} in X is convergence in X if ∥xn∥R −→ 0 at n −→ +∞. Proof. Since in l(E)- valued metric spacs. We say that a sequance xn ⊂ X converges to x ∈ X If for any εl(E) ≻ 0l(E) (where 0l(E) is the zero element in l(E) ) there exists N ∈ N such that for all n > N , dl(E)(xn, x) ⪯ εl(E), then this implies ∥dl(E)(xn, x)∥R < E , E ∈ R. Lemma 5. [2, 9] Suppose that A is a unital C∗-algebra with a unit I: (1) for any x ∈ A+ we have x ⪯ I if and only if ∥x∥ ≤ 1; (2) If a ∈ A+ with ∥a∥ < 1 2 , then I − a is invertable and ∥a(I − a)−1∥ < 1; (3) suppose that a, b ∈ A with a, b ⪰ 0 and ab = ba,then ab ⪰ 0. (4) by Á we denote the set {a ∈ A : ab = ba forall b ∈ A} Let a ∈ Á, if b, c ∈ A with b ⪰ c ⪰ 0 (I − a)−1b ⪰ (I − a)−1c . Definition 20. Let (X, l(E), ∥.∥l(E)) be an l(E) normed space. We call a mapping T : X −→ X is l(E) contractive mapping on X if there exists an M ∈ l(E) with ∥M∥l(E) ≤ 1 such that ∥Tx− Ty∥l(E) ⪯M∗∥x− y∥l(E)M forall x, y ∈ X. Definition 21. An l(E)- Banach space is a complete l(E)-normed space (X, ∥.∥l(E)). Many results on fixed point theorems have been extended from metric spaces to C∗-algebra valued metric spaces with different contraction conditions (see for example [20],[9],[10],[11],[14]) Theorem 1. (Chatterjee Type theorem [1]) Let (X, l(E), ∥.∥l(E)) be an l(E) complete normed space and T : X −→ X be a self mapping satisfy the following contraction condi- tion ∥Tx− Ty∥ l(E) ⪯ M 2 [ ∥Tx− y∥ l(E) + ∥Ty − x∥ l(E) ], where M ∈ (l(E))̀+ with ∥M∥l(E) < 1, Then T has a unique fixed point. Proof. Let x0 ∈ X be arbitrary point and construct a sequence {xn}+∞ n=0 ⊆ X by the way: x1 = Tx0, x2 = Tx1, ....., xn+1 = Txn R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1243 ∥xn+1 − xn∥l(E) = ∥Txn − Txn−1∥l(E) ⪯ M 2 [ ∥Txn − xn−1∥l(E) + ∥Txn−1 − xn∥l(E) ] = M 2 [ ∥xn+1 − xn−1∥l(E) + ∥xn − xn∥l(E) ] ⪯ M 2 [ ∥xn+1 − xn∥l(E) + ∥xn − xn−1∥l(E) ] ⪯ M 2 ∥xn+1 − xn∥l(E) + M 2 ∥xn − xn−1∥l(E) . Thus, (Il(E) − M 2 )∥xn+1 − xn∥l(E) ⪯ M 2 ∥xn − xn−1∥l(E) . Since M ∈ (l(E))̀+ with ∥M 2 ∥l(E) ⪯ 1 2 , one have (Il(E) − M 2 ) −1 ∈ (l(E))̀+, and furthermore M 2 (Il(E) − M 2 ) −1 ∈ (l(E))̀+ with ∥M 2 (Il(E) − M 2 ) −1∥l(E) ≤ 1 . Therefore, ∥xn+1 − xn∥l(E) ⪯ ( M 2 Il(E)−M 2 )∥xn − xn−1∥l(E) ⪯ ( M 2 Il(E)−M 2 )2∥xn−1 − xn−2∥l(E) ... ⪯ ( M 2 Il(E)−M 2 )n∥x1 − x0∥l(E) . Let t = M 2 (Il(E) − M 2 ) −1, B = ∥x1 − x0∥l(E) . Implies ∥xn+1 − xn∥l(E) ⪯ tnB For n+ 1 > m ∥xn+1 − xm∥l(E) ⪯ ∥xn+1 − xn∥l(E) + ∥xn − xn−1∥l(E) + · · ·+ ∥xm+1 − xm∥l(E) ⪯ tnB + tn−1B + · · ·+ tmB ⪯ (tn + tn−1 + · · ·+ tm)B = ∑n k=m t kB = ∑n k=m t k 2 t k 2B 1 2B 1 2 = ∑n k=mB 1 2 t k 2 t k 2B 1 2 = ∑n k=m(t k 2B 1 2 )∗(t k 2B 1 2 ) = ∑n k=m |t k 2B 1 2 |2 R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1244 ⪯ ∥ ∑n k=m |t k 2B 1 2 |2∥l(E)Il(E) ⪯ ∑n k=m ∥B 1 2 ∥2l(E)∥t k 2 ∥2l(E)Il(E) = ∥B∥l(E) ∑n k=m ∥t∥kl(E)Il(E) ⪯ ∥B∥l(E) ∥t∥m l(E) 1−∥t∥m l(E) Il(E) −→ 0l(E)(m −→ +∞), where Il(E) the unite element in l(E), Therefore {xn} is a Cauchy sequence with re- spect to l(E). By the completeness of(X, l(E), ∥.∥l(E)), there exists an x ∈ X such that limn−→+∞ xn = limn−→+∞ Txn−1 = x. Since ∥Tx− x∥l(E) ⪯ ∥Tx− Txn∥l(E) + ∥Txn − x∥l(E) ⪯ M 2 (∥Tx− xn∥l(E) + ∥Txn − x∥l(E)) + ∥Txn − x∥l(E) ⪯ M 2 (∥Tx− x∥l(E) + ∥x− xn∥l(E) + ∥Txn − x∥l(E)) +∥Txn − x∥l(E) = M 2 ∥Tx− x∥l(E) + M 2 ∥x− xn∥l(E) + M 2 ∥Txn − x∥l(E) +∥Txn − x∥l(E). Implies ∥Tx−x∥l(E) ⪯ M 2 Il(E)−M 2 ∥Txn−x∥l(E)+ M 2 Il(E)−M 2 ∥x−xn∥l(E)+ 1 Il(E)−M 2 ∥Txn−x∥l(E) ∥Tx− x∥l(E) ⪯ M 2 Il(E)−M 2 ∥xn+1 − x∥l(E) + 1 Il(E)−M 2 ∥xn+1 − x∥l(E) −→ 0(n −→ +∞) Implies ∥Tx− x∥l(E) = 0 implies Tx = x. To prove the uniquness suppose that y(̸= x) is another fixed point of T, then 0 ⪯ ∥x− y∥l(E) = ∥Tx− Ty∥l(E) ⪯ M 2 (∥Tx− y∥l(E) + ∥Ty − x∥l(E)) Implies ∥x− y∥l(E) ⪯ M 2 Il(E)−M 2 ∥x− y∥l(E) Implies ∥∥x− y∥l(E)∥l(E) ⪯ ∥ M 2 Il(E)−M 2 ∥l(E)∥∥x− y∥l(E)∥l(E) ≺ ∥∥x− y∥l(E)∥l(E) This means that ∥x− y∥l(E) = 0 implies x = y . Therefore the fixed point is unique. R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1245 Theorem 2. (Extension of Chatterjee Type Theorem ) Let (X, l(E), ∥.∥l(E)) be an l(E) complete normed space and T : X −→ X be a self mapping satisfy the following contraction condition ∥Tx− Ty∥l(E) ⪯ M 3 [ ∥x− y∥l(E) + ∥Tx− y∥l(E) + ∥Ty − x∥l(E)], where M ∈ (l(E))̀+ with ∥M∥l(E) < 3 4 , Then T has a unique fixed point. Proof. Le x0 ∈ X be arbitrary point and construct a sequence {xn}+∞ n=0 ⊆ X by the way: x1 = Tx0, x2 = Tx1, ....., xn+1 = Txn. ∥xn+1 − xn∥l(E) = ∥Txn − Txn−1∥l(E) ⪯ M 3 [∥xn−xn−1∥l(E)+∥Txn−xn−1∥l(E)+∥Txn−1−xn∥l(E)] = M 3 [∥xn − xn−1∥l(E) + ∥xn+1 − xn−1∥l(E) + ∥xn − xn∥l(E)] ⪯ M 3 [∥xn − xn−1∥l(E) + ∥xn+1 − xn∥l(E) + ∥xn − xn−1∥l(E)] = M 3 [ 2∥xn − xn−1∥l(E) + ∥xn+1 − xn∥l(E) ] = 2M 3 ∥xn − xn−1∥l(E) + M 3 ∥xn+1 − xn∥l(E) . Thus, (Il(E) − M 3 )∥xn+1 − xn∥l(E) ⪯ 2M 3 ∥xn − xn−1∥l(E). Since M ∈ (l(E))̀+ with ∥M 3 ∥l(E) ≤ 1 4 , one have (Il(E) − M 3 ) −1 ∈ (l(E))̀+, and further- more M 3 (I − M 3 ) −1 ∈ (l(E))̀+ with ∥M 3 (Il(E) − M 3 ) −1∥l(E) ≤ 1 2 ,we have that ∥2(M3 (Il(E) − M 3 ) −1)∥l(E) ≤ 1 . Therefore, ∥xn+1 − xn∥l(E) ⪯ 2( M 3 Il(E)−M 3 )∥xn − xn−1∥l(E) = t∥xn − xn−1∥l(E) ⪯ t2∥xn−1 − xn−2∥l(E) ... ⪯ tn∥x1 − x0∥l(E), where t = 2(M3 (Il(E) − M 3 ) −1). For n+ 1 > m . R. A. Rashwan et al. / Eur. J. Pure Appl. Math, 14 (4) (2021), 1237-1248 1246 ∥xn+1 − xm∥l(E) ⪯ ∥xn+1 − xn∥l(E) + ∥xn − xn−1∥l(E) + · · ·+ ∥xm+1 − xm∥l(E) ⪯ (tn + tn−1 + · · ·+ tm)∥x1 − x0∥l(E). Let B = ∥x1 − x0∥l(E) ⇒ ∥xn+1 − xm∥l(E) = ∑n k=m t kB = ∑n k=m t k 2 t k 2B 1 2B 1 2 = ∑n k=mB 1 2 t k 2 t k 2B 1 2 = ∑n k=m(t k 2B 1 2 )∗(t k 2B 1 2 ) = ∑n k=m |t k 2B 1 2 |2 ⪯ ∥ ∑n k=m |t k 2B 1 2 |2∥l(E)Il(E) ⪯ ∑n k=m ∥B 1 2 ∥2l(E)∥t k 2 ∥2l(E)Il(E) = ∥B∥l(E) ∑n k=m ∥t∥kl(E)Il(E) ⪯ ∥B∥l(E) ∥t∥m l(E) 1−∥t∥m l(E) Il(E) −→ 0l(E)(m −→ +∞), where Il(E) the unite element in l(E), Therefore {xn} is a Cauchy sequence with re- spect to l(E). By the completeness of(X, l(E), ∥.∥l(E)), there exists an x ∈ X such that limn−→+∞ xn = limn−→+∞ Txn−1 = x. Since ∥Tx− x∥l(E) ⪯ ∥Tx− Txn∥l(E) + ∥Txn − x∥l(E) ⪯ M 3 (∥x−xn∥l(E)+∥Tx−xn∥l(E)+∥Txn−x∥l(E))+∥Txn−x∥l(E) ⪯ M 3 (∥x−xn∥l(E)+∥Tx−xn∥l(E)+∥xn+1−x∥l(E))+∥Txn−x∥l(E). Implies ∥Tx−x∥l(E) ⪯ M 3 Il(E)−M 3 (2∥x−xn∥l(E)+∥xn+1−x∥l(E))+ 1 Il(E)−M 3 ∥xn+1−x∥l(E) −→ 0(at n −→ +∞). Then this implies that Tx = x i.e., x is fixed point of T . To prove the uniquencess suppose that y(̸= x) is another fixed point of T, then 0 ≤ ∥x− y∥l(E) = ∥Tx− Ty∥l(E) REFERENCES 1247 ⪯ M 3 (∥x−y∥l(E)+∥Tx−y∥l(E)+∥Ty−x∥l(E)) ⪯M∥x− y∥l(E), Implies 0 ≤ ∥∥x− y∥∥l(E) ≤ ∥M∥x− y∥∥l(E) < ∥∥x− y∥∥l(E) This is contradiction implies x = y. Therefore the fixed point is unique. 4. Conclusions In this paper, we introduced the notions of metric space valued-operator of Hilbert C∗-module. We define some contraction mapping and prove some fixed point theorems (such as Chatterjee and extension of Chatterjee) for a self mappings T on the Banach space l(E). References [1] Chatterjee, SK. Fixed-point theorems. C. R. Acad. Bulgare Sci. 25, 727-730 (1972). [2] Douglas, RG. Banach Algebra Techniques in Operator Theory. Springer, Berlin (1998). [3] Gilbert helmberg. Introduction to spectral theory in Hilbert space. Technological University Eindhoven . [4] Kadelburg et al, Z . 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