ORTHOGONALITY AND LINEAR MAPPINGS IN BANACH MODULES
ORTHOGONALITY AND LINEAR MAPPINGS IN BANACH MODULES
The Pure and Applied Mathematics. 2015. Nov, 22(4): 343-357
• Received : September 03, 2015
• Accepted : September 05, 2015
• Published : November 30, 2015 PDF e-PUB PubReader PPT Export by style
Article
Author
Metrics
Cited by
TagCloud
SUNGSIK, YUN

Abstract
Using the fixed point method, we prove the Hyers-Ulam stability of lin- ear mappings in Banach modules over a unital C *-algebra and in non-Archimedean Banach modules over a unital C *-algebra associated with the orthogonally Cauchy- Jensen additive functional equation.
Keywords
1. INTRODUCTION AND PRELIMINARIES
Assume that X is a real inner product space and f : X → ℝ is a solution of the orthogonal Cauchy functional equation f ( x + y ) = f ( x ) + f ( y ), < x , y >= 0. By the Pythagorean theorem f ( x ) = ∥ x 2 is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space.
G. Pinsker  characterized orthogonally additive functionals on an inner prod- uct space when the orthogonality is the ordinary one in such spaces. K. Sundaresan  generalized this result to arbitrary Banach spaces equipped with the Birkho®- James orthogonality. The orthogonal Cauchy functional equation PPT Slide
Lager Image
in which ⊥ is an abstract orthogonality relation, was first investigated by S. Gudder and D. Strawther  . They defined ⊥ by a system consisting of five axioms and described the general semi-continuous real-valued solution of conditional Cauchy functional equation. In 1985, J. Rätz [47 ] introduced a new definition of orthogonal- ity by using more restrictive axioms than of S. Gudder and D. Strawther. Moreover, he investigated the structure of orthogonally additive mappings. J. Rätz and Gy. Szabó  investigated the problem in a rather more general framework.
Let us recall the orthogonality in the sense of J. Rätz; cf.  .
Suppose X is a real vector space (algebraic module) with dim X ≥ 2 and ⊥ is a binary relation on X with the following properties:
( O 1 ) totality of ⊥ for zero: x ⊥ 0, 0 ⊥ x for all x X ;
( O 2 ) independence: if x , y X - {0}, x y , then x , y are linearly independent;
( O 3 ) homogeneity: if x , y X , x y , then αx βy for all α , β ∈ ℝ;
( O 4 ) the Thalesian property: if P is a 2-dimensional subspace of X , x P and λ ∈ ℝ + , which is the set of nonnegative real numbers, then there exists y 0 P such that x y 0 and x + y 0 λx - y 0 .
The pair ( X ,⊥) is called an orthogonality space (module). By an orthogonality normed space (normed module) we mean an orthogonality space (module) having a normed (normed module) structure.
Assume that if A is a C *-algebra and X is a module over A and if x , y X , x y , then ax by for all a , b A .
Some interesting examples are
(i) The trivial orthogonality on a vector space X defined by ( O 1 ), and for non-zero elements x , y X , x y if and only if x , y are linearly independent.
(ii) The ordinary orthogonality on an inner product space ( X , <., .>) given by x y if and only if < x , y > = 0.
(iii) The Birkhoff-James orthogonality on a normed space ( X , ∥.∥) defined by x y if and only if ∥ x + λy ∥ ≥∥ x ∥ for all λ ∈ ℝ.
The relation ⊥ is called symmetric if x y implies that y x for all x , y X . Clearly examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see   , [7 , 14 , 23 , 24 , 36] ).
The stability problem of functional equations was originated from the following question of Ulam  : Under what condition does there is an additive mapping near an approximately additive mapping? In 1941, Hyers  gave a partial affrmative answer to the question of Ulam in the context of Banach spaces. In 1978, Th.M. Rassias  extended the theorem of Hyers by considering the unbounded Cauchy difference ∥ f ( x + y ) − f ( x ) − f ( y )∥ ≤ ε (∥ x p + ∥ y p ), ( ε > 0, p ∈ [0, 1)). The result of Rassias has provided a lot of infiuence in the development of what we now call generalized Hyers-Ulam stability or Hyers-Ulam stability of functional equations. During the last decades several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The reader is referred to [11 , 21 , 25 , 46] and references therein for detailed information on stability of functional equations.
R. Ger and J. Sikorska  investigated the orthogonal stability of the Cauchy functional equation f ( x + y ) = f ( x ) + f ( y ), namely, they showed that if f is a mapping from an orthogonality space X into a real Banach space Y and ∥ f ( x + y ) − f ( x ) − f ( y )∥ ≤ ε for all x , y X with x y and some ε > 0, then there exists exactly one orthogonally additive mapping g : X Y such that PPT Slide
Lager Image
for all x X .
The first author treating the stability of the quadratic equation was F. Skof  by proving that if f is a mapping from a normed space X into a Banach space Y satisfying ∥ f ( x + y )+ f ( x y )−2 f ( x )−2 f ( y )∥≤ ε for some ε > 0, then there is a unique quadratic mapping g : X Y such that PPT Slide
Lager Image
. P.W. Cholewa  extended the Skof’s theorem by replacing X by an abelian group G . The Skof’s result was later generalized by S. Czerwik  in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [10 , 39] ,   ). PPT Slide
Lager Image
was first investigated by F. Vajzović  when X is a Hilbert space, Y is the scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, H. Drljević  , M. Fochi  , M.S. Moslehian [31 , 32] and Gy. Szabó  generalized this result.
In 1897, Hensel  introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [12 , 27 , 28 , 35] ).
Definition 1.1. By a non-Archimedean field we mean a field 𝕂 equipped with a function (valuation) ｜ · ｜ : 𝕂 → [0,∞) such that for all r , s ∈ 𝕂, the following conditions hold:
(1) ｜ r ｜ = 0 if and only if r = 0;
(2) ｜ rs ｜ = ｜ r s ｜;
(3) ｜ r + s ｜ ≤ max{｜ r ｜, ｜ s ｜}.
Definition 1.2 (  ) . Let X be a vector space over a scalar field 𝕂 with a non- Archimedean non-trivial valuation ｜ · ｜ . A function ∥·∥ : X R is a non- Archimedean norm (valuation) if it satisfies the following conditions:
(1) ∥ x ∥ = 0 if and only if x = 0;
(2) ∥ rx ∥ = ｜ r x ∥ ( r ∈ 𝕂, x X );
(3) The strong triangle inequality (ultrametric); namely,
x + y ∥ ≤ max{∥ x ∥, ∥ y ∥}, x , y X .
Then ( X , ∥.∥) is called a non-Archimedean space.
Assume that if A is a C *-algebra and X is a module over A , which is a non- Archimedean space, and if x , y X , x y , then ax by for all a , b A . Then ( X , ∥.∥) is called an orthogonality non-Archimedean module .
Due to the fact that PPT Slide
Lager Image
Definition 1.3. A sequence { xn }is Cauchy if and only if { x n+1 xn } converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.
Let X be a set. A function d : X × X → [0,∞] is called a generalized metric on X if d satisfies
(1) d ( x , y ) = 0 if and only if x = y ;
(2) d ( x , y ) = d ( y , x ) for all x , y X ;
(3) d ( x , y ) ≤ d ( x , y ) + d ( y , z ) for all x , y , z X .
We recall a fundamental result in fixed point theory.
Theorem 1.4 ( [4 , 13] ) . Let (X, d) be a complete generalized metric space and let J : X X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x X , either PPT Slide
Lager Image
for all nonnegative integers n or there exists a positive integer n0 such that
(1) PPT Slide
Lager Image
, ∀ n n 0 ;
(2) the sequence { J n x } converges to a fixed point y * of J ;
(3) y * is the unique fixed point of J in the set Y = { y X d ( J n0 x , y ) < ∞};
(4) PPT Slide
Lager Image
for all y Y .
In 1996, G. Isac and Th.M. Rassias  were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5 , 6 , 30 , 37 , 38 , 41] ).
This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the orthogonally Cauchy-Jensen additive functional equation in Banach modules over a unital C *-algebra. In Section 3, we prove the Hyers-Ulam stability of the orthogonally Cauchy-Jensen additive functional equation in non-Archimedean Banach modules over a unital C *-algebra.
2. STABILITY OF THE ORTHOGONALLY CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN BANACH MODULES OVER AC*-ALGEBRA
Throughout this section, assume that A is a unital C *-algebra with unit e and unitary group U ( A ) := { u A u * u = uu * = e }, ( X ,⊥) is an orthogonality normed module over A and ( Y , ∥.∥ Y ) is a Banach module over A .
In this section, applying some ideas from [17 , 21] , we deal with the stability problem for the orthogonally Cauchy-Jensen additive functional equation PPT Slide
Lager Image
for all x, y, z X with x y , x z and y z .
Theorem 2.1. Let φ : X 3 → [0,∞) be a function such that there exists an α < 1 with PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . Let f : X Y be a mapping satisfying f(0) = 0 and PPT Slide
Lager Image
for all u U ( A ) and all x , y , z X with x y , x z and y z . If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X.
Proof . Putting y = z = 0 and u = e in (2.2), we get PPT Slide
Lager Image
for all x X , since x ⊥ 0. So PPT Slide
Lager Image
for all x X .
Consider the set PPT Slide
Lager Image
and introduce the generalized metric on S : PPT Slide
Lager Image
where, as usual, inf 𝜙 = +∞. It is easy to show that ( S , d ) is complete (see  ).
Now we consider the linear mapping J : S S such that PPT Slide
Lager Image
for all x X .
Let g , h S be given such that d ( g , h ) = ε . Then PPT Slide
Lager Image
for all x X . Hence PPT Slide
Lager Image
for all x X . So d ( g , h ) = ε implies that d ( Jg , Jh ) ≤α ε . This means that PPT Slide
Lager Image
for all g , h S .
It follows from (2.5) that d ( f , Jf )≤ α .
By Theorem 1.4, there exists a mapping L : X Y satisfying the following:
(1) L is a fixed point of J , i.e., PPT Slide
Lager Image
for all x X . The mapping L is a unique fixed point of J in the set PPT Slide
Lager Image
This implies that L is a unique mapping satisfying (2.6) such that there exists a μ ∈(0,∞) satisfying PPT Slide
Lager Image
for all x X ;
(2) d ( J n f , L ) → 0 as n →∞. This implies the equality PPT Slide
Lager Image
for all x X ;
(3) PPT Slide
Lager Image
, which implies the inequality PPT Slide
Lager Image
This implies that the inequalities (2.3) holds.
Let u = e in (2.2). It follows from (2.1) and (2.2) that PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . So PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . Hence L : X Y is an orthogonally Cauchy-Jensen additive mapping.
Let y = z = 0 in (2.2). It follows from (2.1) and (2.2) that PPT Slide
Lager Image
for all x X . So PPT Slide
Lager Image
for all x X . Hence PPT Slide
Lager Image
for all u U . ( A ) and all x X .
By the same reasoning as in the proof of [ 42 , Theorem], we can show that L : X Y is ℝ-linear, since the mapping f(tx) is continuous in t ∈ ℝ for each x X and L : X Y is additive.
Since L is ℝ-linear and each a A is a finite linear combination of unitary elements (see [ 26 , Theorem 4.1.7]), i.e., PPT Slide
Lager Image
, it follows from (2.7) that PPT Slide
Lager Image
for all x X . It is obvious that PPT Slide
Lager Image
. Thus L : X Y is a unique orthogonally Cauchy-Jensen additive and A -linear mapping satisfying (2.3). PPT Slide
Lager Image
Corollary 2.2. Let 𝜽 be a positive real number and p a real number with 0 < p < 1. Let f : X Y be a mapping satisfying PPT Slide
Lager Image
for all u U ( A ) and all x , y , z X with x y , x z and y z . If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof . The proof follows from Theorem 2.1 by taking φ ( x , y ) = θ (∥ x p +∥ y p +∥ z p ) for all x , y , z X with x y , x z and y z . Then we can choose α = 2 p−1 and we get the desired result. PPT Slide
Lager Image
Theorem 2.3. Let f : X Y be a mapping satisfying (2.2) and f (0) = 0 for which there exists a function φ : X 3 → [0,∞) such that PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof. Let ( S, d ) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping J : S S such that PPT Slide
Lager Image
for all x X .
It follows from (2.4) that d ( f , Jf ) ≤1.
The rest of the proof is similar to the proof of Theorem 2.1. PPT Slide
Lager Image
Corollary 2.4. Let θ be a positive real number and p a real number with p > 1. Let f : X Y be a mapping satisfying (2.8). If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof. The proof follows from Theorem 2.3 by taking PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . Then we can choose α = 2 1−p and we get the desired result. PPT Slide
Lager Image
3. STABILITY OF THE ORTHOGONALLY CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH MODULES OVER AC*-ALGEBRA
Throughout this section, assume that A is a unital C *-algebra with unit e and unitary group U ( A ) := { u A u * u = uu * = e }, ( X ,⊥) is an orthogonality non-Archimedean normed module over A and ( Y , ∥.∥ Y ) is a non-Archimedean Banach module over A . Assume that ｜2｜ ≠ 1.
In this section, applying some ideas from [17 , 21] , we deal with the stability problem for the orthogonally Cauchy-Jensen additive functional equation.
Theorem 3.1. Let φ : X 3 → [0,∞) be a function such that there exists an α < 1 with PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . Let f : X Y be a mapping satisfying f (0) = 0 and (2.2). If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof. It follows from (2.4) that PPT Slide
Lager Image
for all x X .
Let ( S, d ) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping J : S S such that PPT Slide
Lager Image
for all x X .
It follows from (3.3) that d ( f , Jf ) ≤ α .
By Theorem 1.4, there exists a mapping L : X Y satisfying the following:
(1) d ( Jn f , L ) → 0 as n → ∞. This implies the equality PPT Slide
Lager Image
for all x X ;
(2) PPT Slide
Lager Image
, which implies the inequality PPT Slide
Lager Image
This implies that the inequality (3.2) holds.
It follows from (3.1) and (2.2) that PPT Slide
Lager Image
for all u U ( A ) and all x , y , z X with x y , x z and y z . So PPT Slide
Lager Image
for all u U ( A ) and all x , y , z X with x y , x z and y z .
The rest of the proof is similar to the proof of Theorem 2.1. PPT Slide
Lager Image
Corollary 3.2. Let θ be a positive real number and p a real number with p > 1. Let f : X Y be a mapping satisfying (2.8). If for each x X the mapping f ( tx ) is continuous in t ∈ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof . The proof follows from Theorem 3.1 by taking PPT Slide
Lager Image
for all x , y , z X with x y , x z . Then we can choose α =｜2｜ p−1 and we get the desired result. PPT Slide
Lager Image
Theorem 3.3. Let f : X Y be a mapping satisfying (2.2) and f (0) = 0 for which there exists a function φ : X 3 → [0,∞) such that PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof . Let ( S, d ) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping J : S S such that PPT Slide
Lager Image
for all x X .
It follows from (2.4) that d ( f , Jf ) ≤ 1.
The rest of the proof is similar to the proofs of Theorems 2.1 and 3.1. PPT Slide
Lager Image
Corollary 3.4. Let θ be a positive real number and p a real number with 0 < p < 1. Let f : X Y be a mapping satisfying (2.8). If for each x X the mapping f ( tx ) is continuous in t ∈ ℝ, then there exists a unique orthogonally Cauchy-Jensen additive and A-linear mapping L : X Y such that PPT Slide
Lager Image
for all x X .
Proof. The proof follows from Theorem 3.3 by by taking PPT Slide
Lager Image
for all x , y , z X with x y , x z and y z . Then we can choose α = ｜2｜ 1−p and we get the desired result. PPT Slide
Lager Image
Acknowledgements
This research was supported by Hanshin University Research Grant.
References