Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations

Guangyu An; Ying Yao

doi:10.1515/math-2020-0109

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Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations

Guangyu An and Ying Yao

Published/Copyright: December 30, 2020

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$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this paper, we study the Hyers-Ulam-Rassias stability of ( m , n ) -Jordan derivations. As applications, we characterize ( m , n ) -Jordan derivations on C ⁎ -algebras and some non-self-adjoint operator algebras.

Keywords: Hyers-Ulam-Rassias stability; (m,n)-Jordan derivation; C*-algebra

MSC 2010: 16W25; 39B62; 47B47; 47L35

1 Introduction

In 1940, S. Ulam [1] posed a problem about group homomorphisms. Suppose that G 1 is a group, ( G 2 , d ) is a metric group, and ε is a positive number. Does there exist a positive number δ , such that if a mapping f from G 1 into G 2 satisfies the inequality

d ( f ( x y ) , f ( x ) f ( y ) ) ⩽ δ

for each x, y in G 1 , then there exists a homomorphism h from G 1 into G 2 such that

d ( f ( x ) , h ( x ) ) ⩽ ε

for every x in G 1 ? If this problem has a solution, we say that the homomorphisms from G 1 into G 2 are stable.

In 1941, D. Hyers [2] answers the question of Ulam’s problem for Banach spaces. Suppose that X 1 is a normed space and X 2 is a Banach space. If f is a mapping from X 1 into X 2 , and there exists a positive number ε such that

∥ f ( x + y ) − f ( x ) − f ( y ) ∥ < ε

for each x and y in X 1 , then there exists a unique additive mapping h from X 1 into X 2 such that

∥ f ( x ) − h ( x ) ∥ < ε

for every x in X 1 . This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation h ( x + y ) = h ( x ) + h ( y ) .

In 1950, T. Aoki [3] generalized Hyers’s theorem. Suppose that X 1 and X 2 are two Banach spaces. If f is a mapping from X 1 into X 2 , and there exists a positive number ε and 0 ⩽ p < 1 such that

∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ⩽ ε ( ∥ x ∥ p + ∥ y ∥ p )

for each x and y in X 1 , then there exists a positive number θ and a unique linear mapping h from X 1 into X 2 such that

∥ f ( x ) − h ( x ) ∥ ⩽ θ ∥ x ∥ p

for every x in X 1 .

In 1978, Th. Rassias [4] introduced unbounded Cauchy difference and proved the stability of linear mappings between Banach spaces. Suppose that X 1 and X 2 are two Banach spaces. If f is a mapping from X 1 into X 2 , and there exist positive constants ε and 0 ⩽ p < 1 such that

∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ⩽ ϵ ( ∥ x ∥ p + ∥ y ∥ p )

for each x and y in X 1 , then there exists a unique additive mapping h from X 1 into X 2 such that

∥ f ( x ) − h ( x ) ∥ ⩽ 2 ε 2 − 2 p ∥ x ∥ p

for every x in X 1 . Moreover, if f ( t x ) is continuous in t ∈ ℝ for every x in X 1 , then h is a linear mapping, where ℝ denotes the set of the real numbers. In 1991, Z. Gajda [5] proved that the result in [4] is also true when p > 1 , and Z. Gajda [5] also gave an example to show that the Rassias’s stability result is not valid for p = 1 . This phenomenon is called the Hyers-Ulam-Rassias stability.

On the other hand, J. Rassias [6,7,8] generalized Hyers’s stability result by presenting a weaker condition involving a product of different powers of norms. Suppose that X 1 is a normed space and X 2 is a Banach space. If f is a mapping from X 1 into X 2 and there exist positive constants ε ⩾ 0 , p 1 and p 2 in ℝ with p = p 1 + p 2 ≠ 1 , such that

∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ⩽ ϵ ∥ x ∥ p 1 ∥ y ∥ p 2

for each x and y in X 1 , then there exists a unique additive mapping h from X 1 into X 2 such that

∥ f ( x ) − h ( x ) ∥ ⩽ ε | 2 − 2 p | ∥ x ∥ p

for every x in X 1 . Moreover, if f ( t x ) is continuous in t ∈ ℝ for every x in X 1 , then h is a linear mapping.

In this paper, we suppose that A is an algebra over the field of complex numbers ℂ , and all linear mappings are ℂ -linear mappings.

Let ℳ be an A -bimodule. A linear mapping δ from A into ℳ is called a derivation if

δ ( x y ) = x δ ( y ) + δ ( x ) y

for each x, y in A ; and δ is called a Jordan derivation if

δ ( x 2 ) = x δ ( x ) + δ ( x ) x

for every x in A . In 1996, K. Jun and D. Park [9] considered the stability of derivations from a Banach algebra C n [0,1] into a finite dimensional Banach C n [0,1] -bimodule. In 2004, C. Park [10] gave a characterization of the stability of derivations from a Banach algebra into its Banach bimodule.

In 1990, M. Brešar and J. Vukman [11] introduced the concepts of left derivations and Jordan left derivations. Let ℳ be a left A -module. A linear mapping δ from A into ℳ is called a left derivation if

δ ( x y ) = x δ ( y ) + y δ ( x )

for each x, y in A ; and δ is called a Jordan left derivation if

δ ( x 2 ) = 2 x δ ( x )

for every x in A . In 2008, Y. Jung [12] characterized the stability of left derivations.

In 2008, J. Vukman [13] introduced the concept of ( m , n ) -Jordan derivations. Let ℳ be an A -bimodule, m ⩾ 0 and n ⩾ 0 be two fixed integers with m + n ≠ 0 . A linear mapping δ from A into ℳ is called an ( m , n ) -Jordan derivation if

( m + n ) δ ( x 2 ) = 2 m x δ ( x ) + 2 n δ ( x ) x

for every x in A . By simple calculation, it is easy to show that δ is an ( m , n ) -Jordan derivation if and only if

( m + n ) δ ( x y + y x ) = 2 m x δ ( y ) + 2 n δ ( x ) y + 2 m y δ ( x ) + 2 n δ ( y ) x

for each x and y in A . It is clear that the notions of Jordan derivations and Jordan left derivations are particular cases of ( m , n ) -Jordan derivations, obtained when m = n = 1 and when m = 1 and n = 0 , respectively.

This paper is organized as follows. In Section 2, we study the Hyers-Ulam-Rassias stability of ( m , n ) -Jordan derivations.

In Section 3, we give the applications on C ⁎ -algebras and some non-self-adjoint operator algebras.

2 Stability of ( m , n ) -Jordan derivations

In this section, we denote T 1 = { λ ∈ ℂ : | λ | = 1 } . To prove the main theorem, we need the following results.

Lemma 2.1

[14] Suppose that X and Y are two linear spaces. If f is an additive mapping from X into Y such that f ( λ x ) = λ f ( x ) for every x in X and every λ in T 1 , then f is a linear mapping.

In [15], P. Găvruţa generalized the results in [4] and [6] with the admissible control function as follows.

Lemma 2.2

[15] Suppose that ( G , + ) is an abelian group and X is a Banach space. Let φ be a mapping from G × G into [ 0 , ∞ ) such that

φ ˜ ( x , y ) ≔ 2 − 1 ∑ k = 0 ∞ 2 − k φ ( 2 k x , 2 k y ) < ∞

for each x and y in G. If f is a mapping from G into X such that

∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ⩽ φ ( x , y )

for each x and y in G, then there exists a unique additive mapping h from G into X such that

∥ f ( x ) − h ( x ) ∥ ⩽ φ ˜ ( x , x )

for every x in G.

Let A be a normed algebra, ℳ be a Banach A -bimodule, and m , n be two fixed non-negative integers with m + n ≠ 0 . For the sake of convenience, we use the same symbol ∥ ⋅ ∥ to represent the norms on A and ℳ . A mapping φ from A × A into [ 0 , ∞ ) is said to have property ℙ if

(2.1) φ ˜ ( x , y ) ≔ 2 − 1 ∑ k = 0 ∞ 2 − k φ ( 2 k x , 2 k y ) < ∞

for each x , y in A . A mapping f from A into ℳ is said to have property ℚ if f satisfies the following two inequalities:

(2.2) ∥ f ( λ x + y ) − λ f ( x ) − f ( y ) ∥ ⩽ φ ( x , y )

and

(2.3) ∥ ( m + n ) f ( x y + y x ) − 2 m x f ( y ) − 2 n f ( x ) y − 2 m y f ( x ) − 2 n f ( y ) x ∥ ⩽ φ ( x , y )

for each x , y in A and every λ in T 1 . Depending on f, we define a mapping δ f from A into ℳ such that

(2.4) δ f ( x ) = lim k → ∞ 2 − k f ( 2 k x )

for every x in A . The definition of δ f will be used in most of the theorems and corollaries of the paper.

Theorem 2.3

Let A be a normed algebra, ℳ be a Banach A -bimodule, and m , n be two fixed non-negative integers with m + n ≠ 0 . Suppose that φ is a mapping from A × A into [0, ∞ ) satisfying the property ℙ and f is a mapping from A into ℳ satisfying the property ℚ . Then there exists a unique ( m , n ) -Jordan derivation δ f from A into ℳ such that

∥ f ( x ) − δ f ( x ) ∥ ⩽ φ ˜ ( x , x )

for every x in A .

Proof

First we prove that there exists a unique linear mapping δ f from A into ℳ such that

∥ f ( x ) − δ f ( x ) ∥ ⩽ φ ˜ ( x , x )

for every x in A . Let λ = 1 in (2.2), it implies that

(2.5) ∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ⩽ φ ( x , y )

for each x and y in A .

Define a mapping δ f from A into ℳ as in (2.4). By the proof of Lemma 2.2 in [15] and (2.5), and also that the sequence { 2 − k f ( 2 k x ) } is convergent, we know that δ f is a unique additive mapping from A into ℳ such that

∥ f ( x ) − δ f ( x ) ∥ ⩽ φ ˜ ( x , x )

for every x in A . Replacing x, y by 2 k x , 2 k y in (2.2), respectively. It follows that

(2.6) ∥ f ( 2 k ( λ x + y ) ) − λ f ( 2 k x ) − f ( 2 k y ) ∥ ⩽ φ ( 2 k x , 2 k y )

for each x, y in A , and every λ in T 1 . Multiplying 2 − k from the left of (2.6), we can obtain the following inequality

(2.7) ∥ 2 − k f ( 2 k ( λ x + y ) ) − 2 − k λ f ( 2 k x ) − 2 − k f ( 2 k y ) ∥ ⩽ 2 − k φ ( 2 k x , 2 k y )

for each x, y in A , and every λ in T 1 . Taking the limit in (2.7) as k → ∞ and by (2.4), we have that

δ f ( λ x + y ) = λ δ f ( x ) + δ f ( y )

for each x, y in A , and every λ in T 1 . By Lemma 2.1, we know that δ f is a linear mapping.

Next we show that δ f is an ( m , n ) -Jordan derivation. Replacing x, y by 2 k x , 2 k y in (2.3), respectively, we can obtain that

(2.8) ∥ ( m + n ) f ( 2 2 k ( x y + y x ) ) − 2 m 2 k x f ( 2 k y ) − 2 n f ( 2 k x ) 2 k y − 2 m 2 k y f ( 2 k x ) − 2 n f ( 2 k y ) 2 k x ∥ ⩽ φ ( 2 k x , 2 k y )

for each x and y in A . Multiplying 2 − 2 k from the left of (2.8), we have that

(2.9) ∥ ( m + n ) 2 − 2 k f ( 2 2 k ( x y + y x ) ) − 2 m 2 − k x f ( 2 k y ) − 2 n f ( 2 k x ) 2 − k y − 2 m 2 − k y f ( 2 k x ) − 2 n f ( 2 k y ) 2 − k x ∥ ⩽ 2 − 2 k φ ( 2 k x , 2 k y )

for each x and y in A . By the convergence of (2.1), it follows that lim k → ∞ 2 − 2 k φ ( 2 k x , 2 k y ) = 0 . Taking the limit as k → ∞ in (2.9) and by (2.4), we have that

( m + n ) δ f ( x y + y x ) = 2 m x δ f ( y ) + 2 n δ f ( x ) y + 2 m y δ f ( x ) + 2 n δ f ( y ) x

for each x and y in A . It means that δ f is an ( m , n ) -Jordan derivation.□

Corollary 2.4

Let A be a normed algebra, ℳ be a Banach A -bimodule, and m = 1 , n = 1 . Suppose that φ is a mapping from A × A into [0, ∞ ) satisfying the property ℙ and f is a mapping from A into ℳ satisfying the property ℚ . Then there exists a unique Jordan derivation δ f from A into ℳ such that

∥ f ( x ) − δ f ( x ) ∥ ⩽ φ ˜ ( x , x )

for every x in A .

In the following, we assume that A is a unital normed algebra with a unit element e and ℳ is a unital Banach A -bimodule, that is,

e m = m e = m

for every m in ℳ . A mapping φ from A × A into [0, ∞ ) is said to have the property ℙ ˆ if φ satisfies the property ℙ and the following equation:

(2.10) lim k → ∞ 2 − k φ ( 2 k e , x ) = 0

for each x in A .

Lemma 2.5

Let A be a unital normed algebra, ℳ be a unital Banach A -bimodule, and m , n be two fixed non-negative integers with m + n ≠ 0 . Suppose that φ is a mapping from A × A into [0, ∞ ) satisfying the property ℙ ˆ and f is a mapping from A into ℳ satisfying the property ℚ . Then f ( λ x ) = λ f ( x ) for every x in A and λ in ℂ .

Proof

Define a mapping δ f from A into ℳ as in (2.4). By Theorem 2.3, we know that δ f is a unique ( m , n ) -Jordan derivation from A into ℳ such that

(2.11) ∥ f ( x ) − δ f ( x ) ∥ ⩽ 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i x , 2 i x )

for every x in A . Let e be a unit element of A and λ be in ℂ , we have the following inequality:

(2.12) ∥ ( m + n ) δ f ( ( 2 k e ) ( λ x ) ) − λ [ 2 m ( 2 k − 1 e ) f ( x ) + 2 n f ( 2 k − 1 e ) x + 2 m x f ( 2 k − 1 e ) + 2 n f ( x ) ( 2 k − 1 e ) ] ∥ ⩽ | λ | ∥ ( m + n ) f ( 2 k e x ) − 2 m 2 k − 1 e f ( x ) − 2 n f ( 2 k − 1 e ) x − 2 m x f ( 2 k − 1 e ) − 2 n f ( x ) 2 k − 1 e ∥ + | λ | ( m + n ) ∥ δ f ( 2 k e x ) − f ( 2 k e x ) ∥

for every x in A and every k in ℕ . By (2.3), (2.11), and (2.12), it follows that

(2.13) ∥ ( m + n ) δ f ( ( 2 k e ) ( λ x ) ) − λ [ 2 m ( 2 k − 1 e ) f ( x ) + 2 n f ( 2 k − 1 e ) x + 2 m x f ( 2 k − 1 e ) + 2 n f ( x ) ( 2 k − 1 e ) ] ∥ ⩽ | λ | φ ( 2 k − 1 e , x ) + | λ | ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e x , 2 i + k e x )

for every x in A and every k in ℕ . By (2.11) and (2.13), we can obtain the following inequality:

(2.14) ∥ ( m + n ) f ( ( 2 k e ) ( λ x ) ) − λ [ 2 m ( 2 k − 1 e ) f ( x ) + 2 n f ( 2 k − 1 e ) x + 2 m x f ( 2 k − 1 e ) + 2 n f ( x ) ( 2 k − 1 e ) ] ∥ ⩽ ∥ ( m + n ) δ f ( ( 2 k e ) ( λ x ) ) − λ [ 2 m ( 2 k − 1 e ) f ( x ) + 2 n f ( 2 k − 1 e ) x + 2 m x f ( 2 k − 1 e ) + 2 n f ( x ) ( 2 k − 1 e ) ] ∥ + ( m + n ) ∥ f ( ( 2 k e ) ( λ x ) ) − δ f ( ( 2 k e ) ( λ x ) ) ∥ ⩽ | λ | φ ( 2 k − 1 e , x ) + | λ | ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e x , 2 i + k e x ) + ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e λ x , 2 i + k e λ x )

for every x in A and every k in ℕ . Since

(2.15) ∥ 2 m 2 k − 1 ( f ( λ x ) − λ f ( x ) ) ∥ = ∥ 2 m 2 k − 1 e ( f ( λ x ) − λ f ( x ) ) ∥ ⩽ ∥ 2 m 2 k − 1 e f ( λ x ) + 2 n f ( 2 k − 1 e ) λ x + 2 m λ x f ( 2 k − 1 e ) + 2 n f ( λ x ) 2 k − 1 e − ( m + n ) f ( ( 2 k e ) ( λ x ) ) ∥ + ∥ ( m + n ) f ( ( 2 k e ) ( λ x ) ) − 2 m λ 2 k − 1 e f ( x ) − 2 n λ f ( 2 k − 1 e ) x − 2 m λ x f ( 2 k − 1 e ) − 2 n λ f ( x ) 2 k − 1 e ∥ + ∥ 2 n 2 k − 1 ( f ( λ x ) − λ f ( x ) ) ∥

for every x in A and every k in ℕ . By (2.3), (2.14), and (2.15), we can obtain the following inequality:

( 2 m + 2 n ) 2 k − 1 ∥ f ( λ x ) − λ f ( x ) ∥ ⩽ ∥ 2 m 2 k − 1 e f ( λ x ) + 2 n f ( 2 k − 1 e ) λ x + 2 m λ x f ( 2 k − 1 e ) + 2 n f ( λ x ) 2 k − 1 e − ( m + n ) f ( ( 2 k e ) ( λ x ) ) ∥ + ∥ ( m + n ) f ( ( 2 k e ) ( λ x ) ) − 2 m λ 2 k − 1 e f ( x ) − 2 n λ f ( 2 k − 1 e ) x − 2 m λ x f ( 2 k − 1 e ) − 2 n λ f ( x ) 2 k − 1 e ∥ ⩽ φ ( 2 k − 1 e , λ x ) + | λ | φ ( 2 k − 1 e , x ) + | λ | ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e x , 2 i + k e x ) + ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e λ x , 2 i + k e λ x )

for every x in A and every k in ℕ . This means that

(2.16) ( m + n ) ∥ f ( λ x ) − λ f ( x ) ∥ ⩽ 2 − k [ φ ( 2 k − 1 e , λ x ) + | λ | φ ( 2 k − 1 e , x ) + | λ | ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e x , 2 i + k e x ) + ( m + n ) 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k e λ x , 2 i + k e λ x )

for every x in A and every k in ℕ .

Taking the limit in (2.16) as k → ∞ , by the convergence of (2.1) and the property ℙ ˆ , we have that

( m + n ) ( f ( λ x ) − λ f ( x ) ) = 0

for every x in A and λ in ℂ . Since m + n ≠ 0 , f ( λ x ) = λ f ( x ) for every x in A and λ in ℂ .□

The following theorem is the main result in this section.

Theorem 2.6

Proof

Define a mapping δ f from A into ℳ as in (2.4). By Theorem 2.3, we know that δ f is a unique ( m , n ) -Jordan derivation from A into ℳ such that

∥ f ( x ) − δ f ( x ) ∥ ⩽ 2 − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i x , 2 i x )

for every x in A . By Lemma 2.4, it follows that f ( λ x ) = λ f ( x ) for every x in A and λ in ℂ . Hence, we have the following inequality:

(2.17) ∥ f ( x ) − δ f ( x ) ∥ = ∥ 2 − k f ( 2 k x ) − 2 − k δ f ( 2 k x ) ∥ ⩽ 2 − k − 1 ∑ i = 0 ∞ 2 − i φ ( 2 i + k x , 2 i + k x )

for every x in A and every k in ℕ . Taking the limit in (2.17) as k → ∞ and by the convergence of (2.1), we have that f ( x ) = δ f ( x ) for every x in A . It means that f is an ( m , n ) -Jordan derivation.□

By Th. Rassias [4], we have the following result.

Corollary 2.7

Let A be a unital normed algebra, ℳ be a unital Banach A -bimodule, and m, n be two fixed non-negative integers with m + n ≠ 0 . Suppose that f is a mapping from A into ℳ such that

∥ f ( λ x + y ) − λ f ( x ) − f ( y ) ∥ ⩽ θ ( ∥ x ∥ p + ∥ y ∥ p )

and

∥ ( m + n ) f ( x y + y x ) − 2 m x f ( y ) − 2 n f ( x ) y − 2 m y f ( x ) − 2 n f ( y ) x ∥ ⩽ θ ( ∥ x ∥ p + ∥ y ∥ p )

for each x, y in A and every λ in T 1 , where θ ⩾ 0 and 0 < p < 1 . Then f is an ( m , n ) -Jordan derivation.

Proof

Suppose that φ : A × A → [ 0 , ∞ ) is defined by

φ ( x , y ) = θ ( ∥ x ∥ p + ∥ y ∥ p )

for each x, y in A , with θ ≥ 0 and 0 < p < 1 . In the following we show that φ satisfies the property ℙ ˆ . Since 0 < p < 1 , it follows that

(2.18) φ ˜ ( x , y ) ≔ 2 − 1 ∑ k = 0 ∞ 2 − k φ ( 2 k x , 2 k y ) = 2 − 1 ∑ k = 0 ∞ 2 − k θ ( ∥ 2 k x ∥ p + ∥ 2 k y ∥ p ) = 2 − 1 ∑ k = 0 ∞ 2 k ( p − 1 ) θ ( ∥ x ∥ p + ∥ y ∥ p ) < ∞

and

(2.19) lim k → ∞ 2 − k φ ( 2 k e , x ) = lim k → ∞ 2 − k θ ( ∥ 2 k e ∥ p + ∥ x ∥ p ) = lim k → ∞ θ ( 2 k ( p − 1 ) + 2 − k ∥ x ∥ p ) = 0 .

Thus by (2.18), (2.19), and Theorem 2.5, we can conclude that f is an ( m , n ) -Jordan derivation.□

By J. Rassias [6], we have the following result.

Corollary 2.8

∥ f ( λ x + y ) − λ f ( x ) − f ( y ) ∥ ⩽ θ ( ∥ x ∥ p 1 ∥ y ∥ p 2 )

and

∥ ( m + n ) f ( x y + y x ) − 2 m x f ( y ) − 2 n f ( x ) y − 2 m y f ( x ) − 2 n f ( y ) x ∥ ⩽ θ ( ∥ x ∥ p 1 ∥ y ∥ p 2 )

for each x, y in A and every λ in T 1 , where θ ⩾ 0 , and p 1 , p 2 ∈ ℝ with p = p 1 + p 2 < 1 . Then f is an ( m , n ) -Jordan derivation.

Proof

Since p = p 1 + p 2 < 1 , without loss of generality, we can assume that p 1 < 1 . Suppose that φ : A × A → [0, ∞ ) is defined by

φ ( x , y ) = θ ( ∥ x ∥ p 1 ∥ y ∥ p 2 )

for each x, y in A. In the following we show that φ satisfies the property ℙ ˆ . Since p = p 1 + p 2 < 1 and p 1 < 1 , it follows that

(2.20) φ ˜ ( x , y ) ≔ 2 − 1 ∑ k = 0 ∞ 2 − k φ ( 2 k x , 2 k y ) = 2 − 1 ∑ k = 0 ∞ 2 − k θ ( ∥ 2 k x ∥ p 1 ∥ 2 k y ∥ p 2 ) = 2 − 1 ∑ k = 0 ∞ 2 k ( p 1 + p 2 − 1 ) θ ( ∥ x ∥ p 1 ∥ y ∥ p 2 ) < ∞

and

(2.21) lim k → ∞ 2 − k φ ( 2 k e , x ) = lim k → ∞ 2 − k θ ( ∥ 2 k e ∥ p 1 ∥ x ∥ p 2 ) = lim k → ∞ 2 k ( p 1 − 1 ) θ ∥ x ∥ p 2 = 0 .

Thus by (2.20), (2.21) and Theorem 2.5, we can deduce that f is an ( m , n ) -Jordan derivation.□

Corollary 2.9

Let A be a unital normed algebra, ℳ be a unital Banach A -bimodule, and m = 1 , n = 1 . Suppose that φ is a mapping from A × A into [ 0 , ∞ ) satisfying the property ℙ ˆ and f is a mapping from A into ℳ satisfying the property ℚ . Then f is an Jordan derivation.

Remark 1

Suppose that φ is a mapping from A × A into [0, ∞ ) satisfying the property ℙ and f 1 , f 2 are two mappings from A into ℳ satisfying the property ℚ . It is obvious that the mapping | μ | φ is also a mapping from A × A into [0, ∞ ) satisfying the property ℙ , μ f , and f 1 + f 2 are two mappings from A into ℳ satisfying the property ℚ . Moreover, we have that

μ δ f = δ μ f and δ f 1 + δ f 2 = δ f 1 + f 2

are ( m , n ) -Jordan derivations. Let V = { δ f | f : A → ℳ with the property ℚ } and V 0 = { δ f = 0 | f : A → ℳ with the property ℚ } , it follows that W = V / V 0 is a linear space and it is interesting to consider the structure of V 0 and W.

Remark 2

We should notice that if A is an algebra and a mapping φ from A × A into [0, ∞ ) satisfying the property ℙ :

φ ˜ ( x , y ) ≔ 2 − 1 ∑ k = 0 ∞ 2 − k φ ( 2 k x , 2 k y ) < ∞

for each x and y in A , then we cannot deduce the following equation

lim k → ∞ 2 − k φ ( 2 k e , x ) = 0

for each x in A . Indeed, suppose that A = ℝ and define a two-variable non-negative function φ by

φ ( x , y ) = | x y | y 2 + 1

for each x and y in ℝ . It is clear that φ ˜ ( x , y ) is a convergent series for each x , y in ℝ , but lim k → ∞ 2 − k φ ( 2 k , x ) ≠ 0 when x ≠ 0 . It means that (2.10) is not a consequence of (2.1).

Remark 3

In 2010, J. Vukman [16] gave the definition of ( m , n ) -Jordan centralizer. Let A be an algebra and ℳ be an A -bimodule, m ⩾ 0 and n ⩾ 0 be two fixed integers with m + n ≠ 0 . A linear mapping δ from A into ℳ is called an ( m , n ) -Jordan centralizer if

( m + n ) δ ( x 2 ) = m x δ ( x ) + n δ ( x ) x

for every x in A . It is clear that δ is an ( m , n ) -Jordan centralizer if and only if

( m + n ) δ ( x y + y x ) = m x δ ( y ) + n δ ( x ) y + m y δ ( x ) + n δ ( y ) x

for each x and y in A .

Similarly, via the same technique used in the proof of Theorem 2.3, we can characterize the stability of ( m , n ) -Jordan centralizers.

3 Some applications

In [17], G. An and J. He proved that every ( m , n ) -Jordan derivation from a C ⁎ -algebra A into its Banach A -bimodule is zero. Thus, by Theorem 2.5, we have the following result.

Corollary 3.1

Let A be a unital C ⁎ -algebra, ℳ be a unital Banach A -bimodule, and m, n be two fixed non-negative integers with m + n ≠ 0 . Suppose that φ is a mapping from A × A into [0, ∞ ) satisfying the property ℙ ˆ and f is a mapping from A into ℳ satisfying the property ℚ . Then f ≡ 0 .

Let ℋ be a complex Hilbert space, and B ( ℋ ) be the set of all bounded linear operators on ℋ . By a subspace lattice on ℋ , we mean a collection ℒ of subspaces of ℋ with (0) and ℋ in ℒ such that, for every family { M r } of elements of ℒ , both ∩ M r and ∨ M r belong to ℒ , where ∨ M r denotes the closed linear span of { M r } . For a Hilbert space ℋ , we disregard the distinction between a closed subspace and the orthogonal projection onto it. Let ℒ be a subspace lattice on ℋ , ℒ is called a commutative subspace lattice (CSL) if it consists of mutually commuting projections.

Let ℬ be a von Neumann algebra on ℋ , and ℒ ⊆ ℬ be a CSL on ℋ . Then A = ℬ ∩ Alg ℒ is said to be a CSL subalgebra of the von Neumann algebra ℬ .

In [17], G. An and J. He proved that if ℬ is a von Neumann algebra on a Hilbert space ℋ and ℒ ⊆ ℬ is a CSL on ℋ , then every ( m , n ) -Jordan derivation from ℬ ∩ Alg ℒ into B ( ℋ ) is zero. Thus by Theorem 2.5, we have the following result.

Corollary 3.2

Let ℬ be a von Neumann algebra on a Hilbert space ℋ , ℒ ⊆ ℬ be a CSL on ℋ , and m, n be two fixed non-negative integers with m + n ≠ 0 . Suppose that φ is a mapping from ( ℬ ∩ Alg ℒ ) × ( ℬ ∩ Alg ℒ ) into [0, ∞ ) satisfying property ℙ ˆ and f is a mapping from ℬ ∩ Alg ℒ into B ( ℋ ) satisfying the property ℚ . Then f ≡ 0 .

Let A be a unital algebra and ℳ be a unital A -bimodule. Suppose that J is an ideal of A , we say that J is a right separating set (resp. left separating set) of ℳ if for every m in ℳ , J m = { 0 } implies m = 0 (resp. m J = { 0 } implies m = 0 ). We denote by J ( A ) the subalgebra of A generated algebraically by all idempotents in A .

Let ℒ be a subspace lattice on a Hilbert space ℋ , define P ℒ = { E ∈ ℒ : E − ⊉ E } , where E − = ∨ { F ∈ ℒ : F ⊉ E } and let E + = ∩ { F ∈ ℒ : F ⊈ E } . A subspace ℒ is called a completely distributive if L = ∨ { E ∈ ℒ : E − ⊉ L } for every L ∈ ℒ ; ℒ is called a P -subspace lattice if ∨ { E : E ∈ P ℒ } = ℋ or ∩ { E − : E ∈ P ℒ } = ( 0 ) . For some properties of completely distributive subspace lattices and P -subspace lattices, see [19,18]. A totally ordered subspace lattice N is called a nest.

By [18,20], we know that if A and ℳ satisfy one of the following conditions:

A = ℬ ∩ Alg N and ℳ = ℬ , where N is a nest in a factor von Neumann algebra ℬ ;
A = Alg ℒ with (0) + ≠ (0) or ℋ − ≠ ℋ , ℳ = B ( ℋ ) ;
A = Alg ℒ with ∨ { E : E ∈ P ℒ } = ℋ or ∩ { E − : E ∈ P ℒ } = ( 0 ) , ℳ = B ( ℋ ) ;
A = Alg ℒ and ℳ is a dual normal Banach A -bimodule, where ℒ is a completely distributive subspace lattice on a Hilbert space ℋ ;

then ℳ has a right or a left separating set J with J ⊆ J ( A ) .

In [21], G. An and J. Li showed that if A is a unital algebra and ℳ is a unital A -bimodule with a right (left) separating set generated algebraically by idempotents in A , then every ( m , n ) -Jordan derivation from a A into ℳ is zero. By Theorem 2.5, we have the following result.

Corollary 3.3

Let A be a unital algebra, ℳ be a unital A -bimodule, with a right (left) separating set J ⊆ J ( A ) , and m, n be two fixed non-negative integers with m + n ≠ 0 . Suppose that φ is a mapping from A × A into [0, ∞ ) satisfying the property ℙ ˆ and f is a mapping from A into ℳ satisfying the property ℚ . Then f ≡ 0 .

Acknowledgement

The authors thank the reviewers for his or her suggestions. This research was partly supported by the National Natural Science Foundation of China (Grant No. 11801342), the Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ-693), and the Scientific research plan projects of Shaanxi Education Department (Grant No. 19JK0130).

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Received: 2020-05-26

Revised: 2020-10-06

Accepted: 2020-10-26

Published Online: 2020-12-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2020-0109

Keywords for this article

Hyers-Ulam-Rassias stability; (m,n)-Jordan derivation; C*-algebra

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