On the Stability of Quadratic Double Centralizers and Quadratic Multipliers: A Fixed Point Approach

Abstract

We prove the superstability of quadratic double centralizers and of quadratic multipliers on Banach algebras by fixed point methods. These results show that we can remove the conditions of being weakly commutative and weakly without order which are used in the work of M. E. Gordji et al. (2011) for Banach algebras.

1. Introduction

In 1940, Ulam [1] raised the following question concerning stability of group homomorphisms: under what condition does there exist an additive mapping near an approximately additive mapping? Hyers [2] answered the problem of Ulam for Banach spaces. He showed that for two Banach spaces and , if and such that

(1.1)

for all , then there exist a unique additive mapping such that

(1.2)

The work has been extended to quadratic functional equations. Consider to be a mapping such that is continuous in , for all . Assume that there exist constants and such that

(1.3)

Th. M. Rassias in [3] showed with the above conditions for , there exists a unique -linear mapping such that

(1.4)

Găvruţa then generalized the Rassias's result in [4].

A square norm on an inner product space satisfies the important parallelogram equality

(1.5)

Recall that the functional equation

(1.6)

is called quadratic functional equation. In addition, every solution of functional eqaution (1.6) is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings , where is a normed space and is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain is replaced by an abelian group. Indeed, Czerwik in [7] proved the Cauchy-Rassias stability of the quadratic functional equation. Since then, the stability problems of various functional equation have been extensively investigated by a number of authors (e.g, [8–13]).

One should remember that the functional equation is called stable if any approximately solution to the functional equation is near to a true solution of that functional equation, and is supersuperstable if every approximately solution is an exact solution of it (see [14]). Recently, the first and third authors in [15] investigated the stability of quadratic double centralizer: the maps which are quadratic and double centralizer. Later, Eshaghi Gordji et al. introduced a new concept of the quadratic double centralizer and the quadratic multipliers in [16], and established the stability of quadratic double centralizer and quadratic multipliers on Banach algebras. They also established the superstability for those which are weakly commutative and weakly without order. In this paper, we show that the hypothesis on Banach algebras being weakly commutative and weakly without order in [16] can be eliminated, and prove the superstability of quadratic double centralizers and quadratic multipliers on a Banach algebra by a method of fixed point.

2. Stability of Quadratic Double Centralizers

A linear mapping is said to be left centralizer on if , for all . Similarly, a linear mapping satisfying , for all is called right centralizer on . A double centralizer on is a pair , where is a left centralizer, is a right centralizer and , for all . An operator is said to be a multiplier if , for all .

Throughout this paper, let be a complex Banach algebra. Recall that a mapping is a quadratic left centralizer if is a quadratic homogeneous mapping, that is is quadratic and , for all and , and , for all . A mapping is a quadratic right centralizer if is a quadratic homogeneous mapping and , for all . Also, a quadratic double centralizer of an algebra is a pair where is a quadratic left centralizer, is a quadratic right centralizer and , for all (see [16] for details).

It is proven in [8]; that for the vector spaces and and the fixed positive integer , the map is quadratic if and only if the following equality holds:

(2.1)

We thus can show that is quadratic if and only if for a fixed positive integer , the following equality holds:

(2.2)

Before proceeding to the main results, we will state the following theorem which is useful to our purpose.

Theorem 2.1 (The alternative of fixed point [17]).

Suppose that we are given a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then for each given , either , for all , or else exists a natural number such that

(1), for all ,

(2)the sequence is convergent to a fixed point of ,

(3) is the unique fixed point of in the set ,

(4), for all .

Theorem 2.2.

Let be continuous mappings with (), and let be continuous in the first and second variables such that

(2.3)

for all and, for all . If there exists a constant , such that

(2.4)

for all , then there exists a unique double quadratic centralizer on satisfying

(2.5)

(2.6)

for all .

Proof.

From (2.4), it follows that

(2.7)

for all . Putting and replacing by in (2.3), we get

(2.8)

for all . By the above inequality, we have

(2.9)

for all . Consider the set and introduce the generalized metric on :

(2.10)

It is easy to show that is complete. Now, we define the linear mapping by

(2.11)

for all . Given , let be an arbitrary constant with , that is

(2.12)

for all . Substituting by in the inequality (2.12) and using (2.4) and (2.11), we have

(2.13)

for all . Hence, . Therefore, we conclude that , for all . It follows from (2.9) that

(2.14)

By Theorem 2.1, has a unique fixed point in the set . On the other hand,

(2.15)

for all . By Theorem 2.1 and (2.14), we obtain

(2.16)

that is, the inequality (2.5) is true, for all . Now, substitute and by and respectively, put and in (2.15). Dividing both sides of the resulting inequality by , and letting goes to infinity, it follows from (2.7) and (2.3) that

(2.17)

for all and . Putting in (2.17) we have

(2.18)

for all . Hence is a quadratic mapping.

Letting in (2.17), we get , for all and . We can show from (2.18) that for any rational number . It follows from the continuity of and that for each , . So,

(2.19)

for all and . Therefore, is quadratic homogeneous. Putting , in (2.3) and replacing by , we obtain

(2.20)

By (2.7), the right hand side of the above inequality tends to zero as . It follows from (2.15) that , for all . Therefore is a quadratic left centralizer. Also, one can show that there exists a unique mapping which satisfies

(2.21)

for all . The same manner could be used to show that is a quadratic right centralizer. If we substitute and by and in (2.3) respectively, and put , and divide both sides of the obtained inequality by , then we get

(2.22)

Passing to the limit as , and again from (2.7), we conclude that , for all . Therefore is a quadratic double centralizer on . This completes the proof of this theorem.

Now, we establish the superstability of double quadratic centralizers on Banach algebras as follows

Corollary 2.3.

Let with , let be continuous mappings with (), and let

(2.23)

for all and, for all . Then is a double quadratic centralizer on .

Proof.

The result follows from Theorem 2.2 by putting .

3. Stability of Quadratic Multipliers

Assume that is a complex Banach algebra. Recall that a mapping is a quadratic multiplier if is a quadratic homogeneous mapping, and , for all (see [16]). We investigate the stability of quadratic multipliers.

Theorem 3.1.

Let be a continuous mapping with and let be a function which is continuous in the first and second variables such that

(3.1)

for all and all . Suppose exists a constant , , such that

(3.2)

for all . Then there exists a unique multiplier on satisfying

(3.3)

for all .

Proof.

It follows from (3.2) that

(3.4)

for all . Putting , in (3.1), we obtain

(3.5)

for all . Thus

(3.6)

for all . Now we set and introduce the generalized metric on as

(3.7)

It is easy to show that is complete. Consider the mapping defined by , for all . By the same reasoning as in the proof of Theorem 2.2, is strictly contractive on . It follows from (3.6) that . By Theorem 2.1, has a unique fixed point in the set . Let be the fixed point of . Then is the unique mapping with , for all such that there exists satisfying

(3.8)

for all . On the other hand, we have . Thus

(3.9)

for all . Hence

(3.10)

This implies the inequality (3.3). It follows from (3.1), (3.4) and (3.9) that

(3.11)

for all . Thus

(3.12)

for all and . Letting in (3.14), we have , for all and . Now, it follows from the proof of Theorem 2.1 and continuity of and that is -linear. If we substitute and by and in (3.1), respectively, and put and we divide the both sides of the obtained inequality by , we get

(3.13)

Passing to the limit as , and from (3.4) we conclude that , for all .

Using Theorem 3.1, we establish the superstability of quadratic multipliers on Banach algebras.

Corollary 3.2.

Let with , and be a continuous mapping with , and let

(3.14)

for all and, for all . Then is a quadratic multiplier on .

Proof.

The results follows from Theorem 3.1 by putting .