International Journal of Analysis and Applications Volume 18, Number 2 (2020), 194-211 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-194 NONLINEAR (m,p)-ISOMETRIC AND (2,p)-CONCAVE MAPPINGS ON COMPLEX NORMED SPACES EL MOCTAR OULD BEIBA1, AYDAH MOHAMMED AYED AL-AHMADI2,∗ 1Department of Mathematics and Computer Sciences, Faculty of Sciences and Techniques, University of Nouakchott Al Aasriya, P.O. Box 5026, Nouakchott, Mauritania 2Mathematics Departement, College of Sciences and Arts, in Al-Qurayyat, Jouf University, Saudi Arabia ∗Corresponding author: aydahahmadi2011@gmail.com Abstract. Let S be a self mapping on a complex normed space X. In this paper, we study the class of mappings satisfying the following condition ∑ 0≤k≤m (−1)m−k (m k )∥∥Skx − Sky∥∥p = 0, for all x, y ∈ X, where m is a positive integer. We prove some of the properties of these classes of mappings. 1. Introduction Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. In the 1990s, Agler and Stankus [1] studied the following operator. For an operator T ∈ B(H) and a positive integer m, define Bm(T) := m∑ k=0 (−1)m−k ( m k ) T∗kTk. (1.1) An operator T ∈ B(H) is said to be m-contractive ( respectively, m-expansive and m-isometric ) if Bm(T) ≤ 0 ( respectively, Bm(T) ≤ 0 and Bm(T) = 0 ) for some positive integer m. Clearly, T is an Received 2019-12-12; accepted 2020-01-13; published 2020-03-02. 2010 Mathematics Subject Classification. 54E40, 62H86. Key words and phrases. (m, p)-isometries; m-isometric operators; expansive and contractive operator. c©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 194 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-194 Int. J. Anal. Appl. 18 (2) (2020) 195 m-contractive if and only if m∑ k=0 (−1)m−k ( m k )∥∥Tkh‖2 ≥ 0, ∀ h ∈H, T is an m-expansive if and only if m∑ k=0 (−1)m−k ( m k )∥∥Tkh‖2 ≤ 0, ∀ h ∈H, and T is an m-isometry if and only if m∑ k=0 (−1)m−k ( m k )∥∥Tkh‖2 = 0, ∀ h ∈H. Agler and Stankus [1–3] developed a theory for m-isometric operators with rich connections to Toeplitz op- erators, classical function theory and nonstationary stochastic processes. The topics related to m-isometries are currently being studied intensively (see e.g., [4, 17, 22]). In [5, 11, 14, 20] certain types of operators (composition, multiplication, shift) were considered and some conditions under which these operators are m-isometries were given; Let L2(T) be the set of square integrable measurable functions on of the unit circle T = ∂D and H2(T) be the corresponding Hardy space. Let L∞(T) be the set of bounded measurable functions on T and let H∞(T) := L∞∩H2(H). For φ in L∞(T) of the unit circle T = ∂D, the Toeplitz operator Tφ with symbol φ on the Hardy space H2(T) is given by Tφf := P(φf) , (f ∈ H2(T)), where P denotes the orthogonal projection of L2(T) onto H2(T). For φ ∈ L∞(T), a Toeplitz operator Tφ is m-expansive if and only if m∑ j=0 (−1)m−j ( m j )∥∥Tjφk∥∥2 ≤ 0, ∀ k ∈ H2(T), is m-contractive if and only if m∑ j=0 (−1)m−j ( m j )∥∥Tjφk∥∥2 ≥ 0, ∀ k ∈ H2(T), and m-isometric if and only if m∑ j=0 (−1)m−j ( m j )∥∥Tjφk∥∥2 = 0, ∀ k ∈ H2(T). A generalization of m-isometries to operators on general Banach spaces has been presented by several authors in the last years. F. Bayart introduces in [7] the notion of (m,p)-isometries on general (real or complex) Banach spaces. An operator T on a Banach space X is called an (m,p)-isometry if there exists an integer m ≥ 1 and a p ∈ [1,∞), with m∑ k=0 (−1)m−k ( m k ) ‖Tkx‖p = 0 (x ∈H). (1.2) Int. J. Anal. Appl. 18 (2) (2020) 196 Bayart showed that all basic properties of m-isometries on Hilbert spaces (which we should now refer to as (m, 2)-isometries) carry over to (m,p)-isometries on Banach spaces and, further, that (m,p)-isometries are never N-supercyclic if X is of infinite dimension and complex. In [18] the authors took of the restriction p ≥ 1. They considered the equation (1.2) for p > 0 real and studied the role of the second parameter p and also discussed the case p = ∞. Most results in the literature remain valid, with their existing proofs, for p in this extended range. Let X and Y be metric spaces. A mapping S : X −→ Y is called an isometry if it satisfies dY (Sx,Sy) = dX(x,y), for all x,y ∈ X, where dX(., .) and dY (., .) denote the metrics in the spaces X and Y , respectively. In the paper [9], the authors introduced the concept of (m,q)-isometry for maps on a metric space (X,dX) as : a mapping S : X → X is called an (m,q)-isometry for (m ≥ 1, integer, q > 0 real) if it satisfies m∑ k=0 (−1)k ( m k ) dX ( Sm−kx,Sm−ky )q = 0, ∀ x,y ∈ X. In [12] the author consider A(m,p)-isometries, where, for an operator A ∈ B(X), T ∈ B(X) (the algebra of bounded linear operators) is A(m,p)-isometric if β(p)m (T,A,x) := m∑ k=0 (−1)m−k ( m k ) ‖ATkx‖p = 0 (x ∈X). (1.3) Evidently, an I(m,p)-isometry is an (m,p)-isometry; if X = H is a Hilbert space, then m∑ k=0 (−1)m−k ( m k ) ‖ATkx‖p = 0 ⇐⇒ m∑ k=0 (−1)m−k ( m k ) ‖|A|Tkx‖p = 0 (x ∈X). If β (p) m (T,A,x) ≤ 0 ( resp. β (p) m (T,A,x) ≥ 0 ) ,∀ x ∈ X , T is said to be (A,m,p)-expansive ( resp. (A,m,p)- contractive ) . We refer the interested reader to [13, 19] for complete details. A mapping S (not necessarily linear) on a normed space X ( [16]) is an (m,p)-isometry ( m ≥ 1 integer and p > 0 real) if, for all x,y ∈X , ∆pm(S; x,y) := m∑ k=0 (−1)m−k ( m k )∥∥Skx−Sky∥∥p = 0. (1.4) When m = 1, (1.4) is equivalent to ‖Sx−Sy‖ = ‖x−y‖, ∀ x,y ∈X , and when m = 2, (1.4) is equivalent to ‖S2x−S2y‖p − 2‖Sx−Sy‖p + ‖x−y‖p = 0, ∀ x,y ∈X . In [15] it was observed that a sequence ( an ) n∈N in R is concave if an+2 − 2an+1 + an ≤ 0, ∀n ∈ N (1.5) Int. J. Anal. Appl. 18 (2) (2020) 197 or equivalently if an ≥ an−1 + an+1 2 , ∀n ∈ N. (1.6) A sequence ( an ) n∈N in R is log concave if a2n ≥ an−1an+1, ∀n ∈ N (1.7) Remark 1.1. (1) From ( 1.6 ) , we get that any concave sequence of non negative numbers is log concave. (2) If (an)n∈N is a log concave sequence of non negative numbers then an+1 an ≤ an an−1 , ∀n ∈ N. (1.8) Therefore the sequence an+1 an is decreasing sequence of non negative numbers. After a short introduction and some connections with known results in this context, the main results of the paper are presented as follows. In Section 2, we will introduce and study some properties of (2,p)- concave mappings. The main results of this section are Proposition 2.1, Proposition 2.3, Proposition 2.4 and Corollary 2.2. In Section,3, a parallel study of the classes of nonlinear (m,p)-isometric mappings are presented. Exactly we will give conditions under which:a self mapping S is (m,p)-isometry it becomes (k,p))-isometry for 1 ≤ k ≤ m − 1 (Proposition 3.1, Proposition 3.2 and Corollary 3.1. The product of a (m,p)-isometry and a (k,p)-isometry is a (m + k − 1,p)-isometry for k = 1, 2, 3. A power of (2,p)-isometry is again an (2,p)-isometry.. 2. (2,p)-Concave mappings In this section, let (X ,‖.‖) be a complex normad space, S : X −→X is a map. Definition 2.1. Let S be a self map (not necessary linear) on complex normed space §. S is said to be a (2,p)-concave if S satisfy ∥∥S2x−S2y‖p − 2∥∥Sx−Sy∥∥p + ∥∥x−y∥∥p ≤ 0, for all x,y ∈X . Remark 2.1. It is easy to check that a self map S on complex normed space X is (2,p)-concave if and only if the sequence an = ∥∥Snx−Sny∥∥p is concave. Note the following proposition, which lists some general properties of (2,p)-concave mappings. Int. J. Anal. Appl. 18 (2) (2020) 198 Proposition 2.1. Let S be a self map on complex normed space X. If S is an (2,p)-concave. Then the following statements hold: (1) ∥∥Snx−Sny∥∥p + (n− 1)∥∥x−y∥∥p ≤ n.∥∥Sx−Sy∥∥p, x,y ∈X , n = 0, 1, 2, ... (2) ∥∥Sx−Sy∥∥p ≥ n− 1 n ∥∥x−y∥∥p, n ≥ 1, x,y ∈X . (3) ∥∥Sx−Sy∥∥p ≥ ∥∥x−y∥∥p for all x,y ∈X . (4) ∥∥Sx−Sy∥∥ ≤ 2 1p ∥∥x−y∥∥ ∀ x,y ∈ S(X) (the range of S). (5) S is injective. Proof. (1) From the assumption that S is a (2,p)-concave, we get ∥∥S2x−S2y∥∥p −∥∥Sx−Sy∥∥p ≤ ∥∥Sx−Sy∥∥p −∥∥x−y∥∥p. Replacing x by Skx and y by Sky leads to ∥∥Sk+2x−Sk+2y∣∣p −∥∥Sk+1x−Sk+1y∥∥p ≤ ∥∥Sk+1x−Sk+1y∥∥p −∥∥Skx−Sky∥∥p, for k ≥ 0. Thus means that ∥∥Snx−Sny∥∥p = ∑ 1≤k≤n ∥∥Skx−Sky∥∥p −(‖Sk−1x−Sk−1y∥∥p) + ∥∥x−y∥∥p ≤ n (∥∥Sx−Sy∥∥p −∥∥x−y∥∥p) + ∥∥x−y∥∥p ≤ n ∥∥Sx−Sy∥∥p + (1 −n)∥∥x−y∥∥p. Hence, ∥∥Snx−Sny∥∥p + (n− 1))∥∥x−y∥∥p ≤ ∥∥Sx−Sy∥∥p. (2) Form the inequality in (1), it follows ∥∥Sx−Sy∥∥p ≥ n− 1 n ∥∥x−y∥∥p. (3) By taking n →∞ we get ∥∥Sx−Sy∥∥p ≥ ∥∥x−y∥∥p. (4) From the fact that S is a (2,p)-concave we get ∥∥S2x−S2y∥∥p ≤ 2∥∥Sx−Sy∥∥p −∥∥x−y∥∥p ≤ 2∥∥Sx−Sy∥∥p, ∀ x,y ∈X . Int. J. Anal. Appl. 18 (2) (2020) 199 This means that ∥∥S2x−S2y∥∥p ≤ 2 1p ∥∥Sx−Sy∥∥p, or equivalently ∥∥Sx−Sy∥∥p ≤ 2 1p ∥∥x−y∥∥p, ∀ x,y ∈ S(X). (5) The injectivity of S follows immediately from the statement (3). � Remark 2.2. From the injectivity of a (2,p)-concave map S, we get, by the assertion (2) of the Remark 1.1, that for two elements of X x 6= y the sequence (∥∥Sn+1x−Sn+1y∥∥p∥∥Snx−Sny∥∥p ) n is decreasing. Corollary 2.1. Let S be (2,p)-concave on the normed space X and let S−1 be the inverse map of S as a bijection from X on S(X) (the range of S). Then, both S and S−1 are continuous on S(X). Proof. The continuity of S and of S−1 on S(X) follow, respectively, from the assertions (4) and (3) in Proposition 2.1. � Proposition 2.2. Let S be (2, p)-concave on the normed space X and R be an isometric mapping of X such that S and R commute. Then, RS is a (2, p)-concave mapping on X. Proof. The statement follow immediately from the fact ∥∥(RS)2x− (RS)2y∥∥ = ∥∥S2x−S2y∥∥ and ∥∥(RS)x− (RS)y∥∥ = ∥∥Sx−Sy∥∥, ∀ x,y ∈X . � The proof of the following proposition is taken from the statements (1) and (2) of Remark 1.1. For the reader ’s convenience. here we give a proof. Proposition 2.3. Let S be a self map on complex normed space X such is a (2,p)-concave..Then the following assertions holds: (1) ∥∥Sx−Sy∥∥2p ≥ ∥∥x−y∥∥p∥∥S2x−S2y∥∥p for all x,y ∈X . (2) For each n and x,y ∈ X such that x 6= y,the sequence (∥∥Sn+1x−Sn+1y∥∥p∥∥Tnx−Tny∥∥p ) n≥0 is monotonically decreasing to 1. Int. J. Anal. Appl. 18 (2) (2020) 200 Proof. (1) Since S is a (2,p)-concave map, it follows d ∥∥Sx−Sy∥∥2p ≥ (∥∥x−y∥∥p + ∥∥S2x−S2y∥∥p 2 )2 ≥ (∥∥x−y∥∥p2 ∥∥S2x−S2y∥∥p2 )2 ≥ ∥∥x−y∥∥p∥∥S2x−S2y∥∥p, ∀ x,y ∈X . (2) By Observing that the (2,p)-concavity of S implies that ∥∥Sn+1x−Sn+1y∥∥p − 2∥∥Snx−Sny∥∥p + ∥∥Sn−1x−Sn−1y∥∥p ≤ 0. (2.1) On the other hand, since (∥∥Sn−1x−Sn−1y∥∥p2 −∥∥Sn+1x−Sn+1y∥∥p2 )2 ≥ 0, we obtain ∥∥Sn−1x−Sn−1y∥∥p2 ∥∥Sn+1x−Sn+1y∥∥p2 ≤ ∥∥Sn+1x−Sn+1y∥∥p + d∥∥Sn−1x−Sn−1y∥∥p 2 ≤ ∥∥Snx−Sny∥∥p (by (2.1)). Consequently, ∥∥Sn−1x−Sn−1y∥∥p∥∥Sn+1x−Sn+1y∥∥p ≤ ∥∥Snx−Sny∥∥2p. Hence, ∥∥Sn+1x−Sn+1y∥∥p∥∥Snx−Sny∥∥p ≤ ∥∥Snx−Sny∥∥p∥∥Sn−1x−Sn−1y∥∥p , so the sequence is monotonically decreasing. To calculate its limit in view of the statement (3) of Proposition 2.1, divide (2.1) by ∥∥Sn−1x−Sn−1y∥∥p to get 1 − 2 ∥∥Snx−Sny)p∥∥Sn−1x−Sn−1y∥∥p + ∥∥Sn+1x−Sn+1y∥∥p∥∥Snx−Tny∥∥p ∥∥Snx−Sny∥∥p∥∥Sn−1x−Sn−1y∥∥p ≤ 0. By tanking n tend to infinity we obtain that ∥∥Snx−Sny∥∥p∥∥Sn−1x−Sn−1y∥∥p −→ 1 as n →∞. � Proposition 2.4. Let S be a self map on a normed space X . If S is an bijective (2,p)-concave, then , S−1 is an (2,p)-concave. Int. J. Anal. Appl. 18 (2) (2020) 201 Proof. Since S is an (2,p)-concave, we have ∥∥S2x−S2y∥∥p − 2∥∥Sx−Sy∥∥p + ∥∥x−y∥∥p ≤ 0, ∀ x,y ∈X . Under the assumption that S is bijective, it follows by replacing x by S−2x and y by S−2y that ∥∥x−y∥∥p − 2∥∥S−1x−S−1y∥∥p + ∥∥S−2x−S−2y∥∥p ≤ 0, ∀ x,y ∈X . Therefore S−1 is a (2,p)-concave. � Corollary 2.2. Let S be a self map on a normed space X . If S is an bijective (2,p)-concave, then , S is isometric mapping. Proof. Since S is an (2,p)-concave , we have in view of the statement (2) of Proposition 2.1 that ∥∥Sx−Sy∥∥p ≥ ∥∥x−y∥∥p. Moreover, since S is a bijective (2,p)-concave, we have S−1 is (2,p)-concave, hence ∥∥S−1u−S−1w∥∥p ≥ ∥∥u−w∥∥p ∀ u,w ∈X . Letting u = Sx and w = Sy, this implies ∥∥Sx−Sy∥∥p = ∥∥x−y∥∥p, for all x,y ∈X . This means that S is isometric. � 3. Some properties of nonlinear (m, p)-isometric mapping In this section, let (X ,‖.‖) be a complex normad space, S : X −→X is a map, m ∈ N and p ∈ (0,∞) is a real number. We define the quantity ∆pm(S; x,y) := ∑ 0≤k≤m (−1)m−k ( m k )∥∥Skx−Sky∥∥p, (3.1) for all x,y ∈X . Definition 3.1. ( [16]) Let S : X −→X be a map (not necessary linear). S is said to be an (m,p)-isometric mapping for some positive integer m and p ∈ (0,∞) if S satisfying ∑ 0≤k≤m (−1)m−k ( m k )∥∥Skx−Sky∥∥p = 0, for all x,y ∈X. Int. J. Anal. Appl. 18 (2) (2020) 202 Remark 3.1. (1) A self mapping S on X is an (1,p)-isometry if ‖x−y‖ = ‖Sx−Sy‖ ∀ x,y ∈X . (2) A self mapping S on X is an (2,p)-isometry if ‖S2x−S2y‖p − 2‖Sx−Sy‖p + ‖x−y‖p = 0, ∀ x,y ∈X . (3) A self mapping S on X is an (3,p)-isometry if for all x,y ∈X , ‖S3x−S3y‖p − 3‖S2x−S2y‖p + 3‖Sx−Sy‖p −‖x−y‖p = 0, ∀ x,y ∈X . Remark 3.2. The following remarks are obvious consequence of Definition 2.1. (1) A (1,p)-isometry is an isometry and vice versa. (2) Every isometric mapping is (m,p)-isometric mapping for all m ≥ 1 and p ∈ (0,∞).. Theorem 3.1. Let S be a self map on complex normed space X. Then following statements hold: (1) ∆pm(S; x,y) = ∆ p m−1(S; x,y) − ∆ p m−1(S; Sx,Sy), (3.2) for all x,y ∈X and m ∈ N. (2) If S is an (m,p)-isometry, then S is an (k,p)-isometry for all integer k with k ≥ m. Proof. (1) In view of the standard formula ( m k ) = ( m−1 k ) + ( m−1 k−1 ) for binomial coefficients, it follows that ∆pm(S; x,y) = ∑ 0≤k≤m (−1)k ( m k )∥∥Skx−Sky‖p = ∥∥x−y∥∥p+ ∑ 1≤k≤m−1 (−1)k ( m k )∥∥Skx−Sk∥∥p+(−1)m∥∥Smx−Smy∥∥p = ∥∥x−y∥∥p+ ∑ 1≤k≤m−1 (−1)k ((m− 1 k ) + ( m− 1 k − 1 )∥∥(Skx−Sky∥∥p + +(−1)m ∥∥Smx−Smy∥∥p = ∆ p m−1(S; x,y) − ∆ p m−1(S; Sx,Sy). (2) The statement (2) follows from the statement (1). � Int. J. Anal. Appl. 18 (2) (2020) 203 Lemma 3.1. Let S be a self map on a complex normed space X, then S is a (2,p)-isometry if and only if ∥∥Skx−Sky∥∥p −k∥∥Sx−Sy‖p + (k − 1)∥∥x−y∥∥p = 0, ∀ k = 0, 1, · · · ,x,y ∈X . Proof. The proof follows by using a mathematical induction, so we omit it. � Following [10], we say that an m-isometry S is strict if m = 1, or m ≥ 2 and S is not an (m− 1)-isometry. Examples of strict m-isometries for any m ≥ 2 are provided in [6, Proposition 8] (see also [23, Example 2.3]). Proposition 3.1. Let S be a self map on a complex normed X that is a (m,p)-isometric and a (2,p)-concave mapping. Then S is a (2,p)-isometric mapping. Proof. Since S is an (m,p)-isometry, ∆pm(S; x,y) = 0 for all x,y ∈X . By (3.2), we have ∆ p m−1(S; x,y) = ∆ p) m−1(S; Sx,Sy). Therefore ∆ p m−1(S; x,y) = ∆ p) m−1(S; Sx,Sy) = ∆ p m−1(S; S nx,Sny); for n = 1, 2, .... From the assumption that S is a (2,p)-concave it follows in view of the statement (2) of Proposition 2.1 that   ∥∥Sx−Sy∥∥p −∥∥x−y∥∥p ≥ 0 ∥∥x−y∥∥p − 2∥∥Sx−Sy)p + ∥∥S2x−T2y∥∥p ≤ 0 This means that the sequence (∥∥Sn+1x − Sn+1y∥∥p −∥∥Snx − Sny∥∥p) n≥0 is monotonically non-increasing and therefore bounded so, is converges. Hence there exists a constant β such that ∥∥Sn+1x−Sn+1y∥∥p −∥∥Snx−Sny∥∥p −→ β as n −→∞. Note that ∆ p m−1(S; S nx,Sny) = ∑ 0≤k≤m−−1 (−1)k ( m− 2 k )[∥∥Sn+kx−Sn+k∥∥p−∥∥Sn+1+kx−Sn+1+ky∥∥p]. Letting n →∞ in the preceding equality leads to ∆ p m−1(S; S nx,Sny) → ∑ 0≤k≤m−2 (−1)k ( m− 2 k ) β = 0. We obtain that ∆ p m−1(S; x,y) = 0. Applying the corresponding results of the (m− 1,p)-isometry and (2.p)- concave mapping, we obtain that ∆ p m−2(S; x,y) = 0. Continue these processes we get ∆ p 2(S; x,y) = 0. Consequently, S is a (2,p)-isometric mapping. � Int. J. Anal. Appl. 18 (2) (2020) 204 Proposition 3.2. Let S be a self mapping on a complex normed space X that is a contractive (∥∥Sx−Sy∥∥ ≤∥∥x−y∥∥, ∀ x,y ∈X). In addition If S is an (m,p)-isometry then S is an (m− 1,p)-isometry for m ≥ 2. Proof. Under the assumption that S is a contractive mapping, it follows that, ∥∥Sn+1x−Sn+1y∥∥p ≤ ∥∥Snx−Sny∥∥p, ∀ x,y ∈X and n ∈ N. This means that (∥∥Snx−Sny∥∥p) n∈N is deceasing sequence, so convergent. From the fact that S is an (m,p)-isometry and together (??), we get ∆ p m−1(S; x,y) = ∆ p m−1(S; Sx,Sy) = ... = ∆ p m−1(S; S nx,Sny). On the other hand, we have ∆ p m−1(S; S nx,Sny) = ∆Pm−2(S; S nx,Sny) − ∆pm−2(S; S n+1x,Sn+1y), so that ∆ p m−1(S; S nx,Sny) = m−2∑ j=0 (−1)j ( m− 2 j )[∥∥Sn+jx−Sn+jy∥∥p −∥∥Sn+1+jx−Sn+1+jy∥∥p]. By taking the limit n →∞ in the preceding equality leads to ∆ p m−1(S; S nx,Sny)to0. Consequently,, ∆ p m−1(S; x,y) = 0 and hence, T is an (m− 1,p)-isometry. � As a consequence of this proposition, we have the following corollary. Corollary 3.1. Let S be a self mapping on a complex normed space X which is a contractive. Then S is an (m,p)-isometry if and only if S is an isometry. Lemma 3.2. Let S be a self mapping on a complex normed space X . Then S is an (2, p)-isometry if and only if ∥∥Snx−Sny∥∥p −∥∥Sn−1x−Sn−1y∥∥p = ∥∥Sx−Sy∥∥p −∥∥x−y∥∥p, for all integer n ≥ 2 and x,y ∈X . Proof. The proof is obvious by mathematical induction. � The author in [21] show that if S is a 2-isometric operator on a Hilbert space, then Sn is a 2-isometric operator. Int. J. Anal. Appl. 18 (2) (2020) 205 Theorem 3.2. Let S be a self map on complex normed space X . if S is an (2,p)-isometry, then so is Tn for all n ∈ N. Proof. We will induct on n, the result obviously holds for n = 1. Suppose then the assertion holds for n ≥ 2, i.e ∥∥S2nx−S2ny∥∥p − 2∥∥Snx−Sny∥∥p + ∥∥x−y∥∥p = 0, ∀x,y ∈X . Then ∥∥S2n+2x−S2n+2y∥∥p − 2∥∥Sn+1x−Sn+1y∣∣p + ∥∥x−y∥∥p = ∥∥S2S2nx−S2S2ny∥∥p − 2∥∥Sn+1x−Sn+1y∥∥p + ∥∥x−y∥∥p = 2 ∥∥S2n+1x−S2n+1y∥∥p −∥∥S2nx−S2ny∥∥p −2 ∥∥Sn+1x−Sn+1y∥∥p + ∥∥x−y∥∥p = 2 ( 2 ∥∥Sn+1x−Sn+1y∥∥p−∥∥Sx−Sy∥∥p)−∥∥S2nx−S2ny∥∥p −2 ( ‖Sn+1x−Sn+1y ∥∥p+∥∥x−y∥∥p = 2 ∥∥Sn+1x−Sn+1y∥∥p −∥∥S2nx−S2ny∥∥p −2 ∥∥Sx−Sy∥∥p + ∥∥x−y∥∥p = 2 ∥∥Sn+1x−Sn+1y∥∥p−(2∥∥Snx−Sny∥∥p−∥∥x−y∥∥p) −2 ∥∥Sx−Syw�p+∥∥x−y∥∥p ≤ 2 ∥∥Sn+1x−Sn+1y∥∥2 − 2∥∥Snx−Sny∥∥p − 2∥∥Sx−Sy∥∥p + 2∥∥‖x−y∥∥p = 2 (∥∥Sx−Sy∥∥p−∥∥x−y∥∥p)−2∥∥Sx−Sy∥∥p + 2∥∥x−y∥∥p (by Lemma3.2) = 0. Thus means that Sn is (2,p)-isometric mapping. � The following theorem gives a characterization of (3,p)-isometric mappings. Theorem 3.3. Let S be a self mapping for a normed space X. Then S is an (3,p)-isometric mapping if and only if S satisfies ∥∥Snx−Sny∥∥p = ∥∥x−y∥∥p + nΨ1(S,x,y) + n2Ψ2(S,x,y) (3.3) where Ψ2(S,x,y) = 1 2 (∥∥S2x−S2y∥∥p − 2∥∥Sx−Sy‖p + ∥∥x−y‖p) (3.4) and Ψ1(S,x,y) = 1 2 ( − ∥∥S2x−S2y∥∥p + 4∥∥Sx−Sy∥∥p − 3∥∥x−y‖p) (3.5) Int. J. Anal. Appl. 18 (2) (2020) 206 Proof. We prove the if part of the theorem. Assume that S satisfies (3.3). For n = 3 we obtain ∥∥S3x−S3y∥∥p = ∥∥x−y∥∥p + 3Ψ1(S,x,y) + 9Ψ2(S,x,y) = ∥∥x−y∥∥p + 3 2 ( − ∥∥S2x−S2y∥∥p + 4∥∥Sx−Sy∥∥p − 3∥∥x−y∥∥p) + 9 2 (∥∥S2x−S2y‖p − 2∥∥Sx−Sy∥∥p + ∥∥x−y∥∥p) = ∥∥x−y‖p − 3∥∥S2x−S2y∥∥p − 3∥∥Sx−Sy∥∥p. It follows that ∥∥S3x−S3y‖p − 3∥∥S2x−S2y∥∥p + 3∥∥Sx−Sy∥∥p −∥∥x−y∥∥p = 0, so that, S is an (3,p)–isometry. We prove the only if part. Assume that S is an (3,p)-isometry. We prove (3.3) by mathematical induction. For n = 1 it is true. Assume that (3.3) is true for n and prove it for n + 1. Indeed, for all x,y ∈X we have ∥∥Sn+1x−Sn+1y∥∥p = ∥∥SnSx−SnSy∥∥p = ∥∥Sx−Sy‖p + nΨ1(S,Sx,Sy) + n2Ψ2(S,Sx,Sy) = ∥∥Sx−Sy∥∥p + n 2 ( − ∥∥S3x−S3y∥∥p + 4∥∥S2x−S2y∥∥p − 3∥∥Sx−Sy∥∥p) + n2 2 (∥∥S3x−S3y∥∥p − 2∥∥S2x−S2y∥∥p + ∥∥Sx−Sy∥∥p) = ( n2 −n 2 )∥∥S3x−S3y∥∥p −(n2 − 2n)∥∥S2 −S2y∥∥p + ( n2 − 3n + 2 2 )∥∥Sx−Sy∥∥p. Now, using the fact that T is an (3,p)-isometry we can obtained ∥∥Sn+1x−Sn+1y∥∥p = ( n2 −n 2 )(∥∥x−y∥∥p + 3∥∥S2x−S2y∥∥p − 3∥∥Sx−Sy∥∥p) + − ( n2 − 2n )∥∥S2x−S2y∥∥p + ( n2 − 3n + 2 2 )∥∥Sx−Sy∥∥p = ( n2 + n 2 )∥∥S2x−s2y∥∥p + (−2n2 + 2 2 )∥∥Sx−Sy∥∥p + ( n2 −n 2 )∥∥x−y∥∥p Int. J. Anal. Appl. 18 (2) (2020) 207 = ( n2 + n 2 )(∥∥x−y∥∥p + 2Ψ1(S,x,y) + 4Ψ2(S,x,y)) + ( −2n2 + 2 2 )(∥∥x−y∥∥p + Ψ1(S,x,y) + Ψ2(S,x,y)) + (n2 −n 2 )∥∥x−y∥∥p = ∥∥x−y∥∥p + (n + 1)Ψ1(S,x,y) + (n + 1)2Ψ2(S,x,y). � Theorem 3.4. Let T,S be a self mappings on a complex normed space X such that TS = ST . The following properties hold for all x,y ∈X. (1) If S is an (1,p)-isometry, then ∆pm(TS; x,y) = ∆ p m(T ; x,y). (3.6) (2) If S is a (2.p)-isometry, then ∆ p m+1(TS; x,y) = (m + 1)∆ p m(T ; TSx,TSy) − (m + 1)∆ p m(T, .Tx,Ty) +∆ p m+1(T ; x,y). (3.7) (3) If S is an ((3,p)-isometry, then ∆m+2(TS; x,y) = ∆m+2(T ; x,y) + (m + 2)(m + 1) 2 [ ∆pm(T; T 2S2x,T2S2y) +∆pm(T ; T 2Sx,T2Sy) + ∆pm(T ; T 2x,T2y) ] + 5(m + 2) 2 ∆m+1(T ; TSx,TSy) − 1 2 ∆m(T ; Tx,Ty). (3.8) Proof. (1) From the fact that S is an (1,p)-isometry,i.e.; ∥∥Skx−Sky∥∥p = ∥∥x−y∥∥p ∀ x,u ∈X , and the fact TS = ST it follows by elementary calculation that ∆pm(TS; x,y) = ∑ 0≤k≤m (−1)m−k ( m k )∥∥(TS)kx− (TS)ky∥∥p = ∑ 0≤k≤m (−1)m−k ( m k )∥∥SkTk −SkTky∥∥p = ∑ 0≤k≤m (−1)m−k ( m k )∥∥Tkx−Tky∥∥p = ∆pm(T; x,y). (2) Assume that S is an (2,p)-isometry. Then we have by using [19, Lemma3.4] that ∥∥Skx−Sky∥∥p = k∥∥Sx−Sy∥∥p + (1 −k)∥∥x−y∥∥p, ∀ x,y ∈X , k = 0, 1, · · · . Int. J. Anal. Appl. 18 (2) (2020) 208 A simple calculation shows that βAm+1(TS; x,y) = ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k )∥∥(TS)kx− (TS)ky∥∥p = ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k )∥∥SkTk −SkTky∥∥p = ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k )[ k ∥∥STkx−STky∥∥p + (1 −k)∥∥Tkx−Tky∥∥p] = ∑ 1≤k≤m+1 (−1)m+1−kk ( m + 1 k )∥∥TkSx−TkSy∥∥p + ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k ) (−k + 1) ∥∥Tkx−Tky∥∥p = (m + 1) ( ∑ 0≤k≤m (−1)m−k ( m k )∥∥Tk+1Sx−Tk+1Sy∥∥p)− (m + 1)∆pm(T ; Tx,Ty) +∆ p m+1(T ; x,y) = (m + 1)∆Am(T ; TSx,TSy) − (m + 1)∆ p m(T ; Tx,Ty) + ∆ p m+1(T ; x,y). (3) Assume that S is an (A, 3)-isometry and TS = ST. In view of Theorem 3.3 we have that βAm+2(TS) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )∥∥(ST)kx− (ST)ky∥∥p = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )∥∥SkTkx−SkTky∥∥p = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )[∥∥Tkx−Tky∥∥p + kΨ1(S,Tkx,Ty) + k2Ψ2(S,Tkx,Ty)] = { ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )∥∥Tkx−Tky∥∥p ︸ ︷︷ ︸ I + ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) kΨ1(S,T kx,Tky) ︸ ︷︷ ︸ J + ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2Ψ2(S,T kx,Tky) ︸ ︷︷ ︸ K } . Int. J. Anal. Appl. 18 (2) (2020) 209 Clearly I = ∆ p m+2(T ; x,y). J = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) kΨ1(S,T kx,Tky) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k 1 2 ( − ∥∥S2Tkx−S2Tky∥∥p + 4∥∥STkx−STky∥∥p − 3∥∥Tkx−Tky‖p) = − 1 2 (m + 2)∆m+1(T ; TS 2x,TS2y) + 2(m + 2)∆m+1(T; TSx,TSy) − 3 2 ∆m+1(T; Tx,Ty). K = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2Ψ2(S,T kx,Tky) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2 1 2 (∥∥S2Tkx−S2Tky∥∥p − 2∥∥STkx−STky‖p + ∥∥Tkx−Tky‖p) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2 1 2 (∥∥TkS2x−TkS2y∥∥p − 2∥∥TkSx−TkSy‖p + ∥∥Tkx−Tky‖p) = (−1)m+1(m + 2) (∥∥TkS2x−TkS2y∥∥p − 2∥∥TkSx−TkSy‖p + ∥∥Tkx−Tky‖p) + ∑ 2≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2 1 2 (∥∥TkS2x−TkS2y∥∥p − 2∥∥TkSx−TkSy‖p + ∥∥Tkx−Tky‖p) By observing that k2 = k(k − 1) + k and k2 ( m + 2 k ) = (m + 2)(m + 1) ( m k − 2 ) + (m + 2) ( m + 1 k − 1 ) , ; k ≥ 2. K = (m + 2)(m + 1) 2 [ ∆pm(T; T 2S2x,T2S2y) + ∆pm(T ; T 2Sx,T2Sy) + ∆pm(T; T 2x,T2y) ] + (m + 2) 2 [ ∆ p m+1(T; TS 2x,TS2y) + ∆ p m+1(T ; TSx,TSy) + ∆ p m+1(T; Tx,Ty) ] . By combining I,J and K we obtain ∆m+2(TS; x,y) = ∆m+2(T ; x,y) + (m + 2)(m + 1) 2 [ ∆pm(T; T 2S2x,T2S2y) + ∆pm(T; T 2Sx,T2Sy) +∆pm(T; T 2x,T2y) ] + 5(m + 2) 2 ∆m+1(T ; TSx,TSy) − 1 2 ∆m(T; Tx,Ty). This completes the proof of the theorem. � The proof of the following corollary follows by Combing Theorem 3.4 and Theorem 3.1. Corollary 3.2. Let T,S be a self mappings on a complex normed space X such that TS = ST . If T is an (m,p)-isometry and S is an (n,p)-isometry for n ∈ {1, 2, 3}, then TS is an (m + n− 1,p)-isometry for n ∈{1, 2, 3}. Int. J. Anal. Appl. 18 (2) (2020) 210 Theorem 3.5. Let S be a self map on a complex normed space X. If S is bijective (m,p)-isometry, then S−1 is an (m,p)-isometry. Proof. Since S is an (m,p)-isometry it follows that ∆pm(S; x,y) = 0; ∀ x,y ∈ X . Taking into account the fact that S is a bijective map we get by direct calculation 0 = ∆pm(S; S −mx,S−my) = ∑ 0≤k≤m (−1)m−k ( m k )∥∥Sk−mx−Sk−my∥∥p = (−1)m∆pm(S −1; x,y). 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