Hyers-Ulam Stability of Differentiation Operator on Hilbert Spaces of Entire Functions

Abstract

We investigate the Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions. We give a necessary and sufficient condition in order that the operator has the Hyers-Ulam stability and also show that the best constant of Hyers-Ulam stability exists.

1. Introduction

In 1940, the first stability problem concerning group homomorphisms was raised by Ulam [1]. Let be a group and let be a metric group with a metric . Given any , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?

In the following years, Hyers affirmatively answered the question of Ulam for the case where and are Banach spaces (see [2]). Furthermore, the result of Hyers has been generalized by Rassias (see [3]).

Since then, the stability of many algebraic, differential, integral, operatorial, functional equations have been extensively investigated (see [4–17] and the references therein).

In this paper, we discuss the Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions and give a necessary and sufficient condition in order that the operator has the Hyers-Ulam stability, and we show that the best constant of Hyers-Ulam stability exists.

2. Hilbert Spaces of Entire Functions

In this section, we describe the Hilbert spaces of entire functions in which the rest of our work is set and record their most basic properties. About the function spaces, we recommend the research papers [18, 19]. For the sake of coherency we recall a few basic definitions, notions, and theorems from [18], and we also give some typical examples; in particular Fock space in these examples is a very important tool for quantum stochastic calculus in the case of quantum probability (see [20–22]).

Let us call an entire function a comparison function if for each , and the sequence of ratios decreases to zero as increases to . For each comparison function , we define to be the Hilbert space of power series for which

It is easy to check that each element of is an entire function and that every sequence convergent in the norm of the space is uniformly convergent on compact subsets of the plane. In this case, the inner product of is given by and the functions form an orthonormal basis for . We can see that the polynomials are dense in .

Example 1. We consider the comparison function ; that is, ; by a simple calculation, we can see that , and then is the famous Fock space.

Example 2. If we put comparison function , that is, , we can see that on .

Throughout this paper, let be the differentiation operator defined by

An important result about is the following theorem.

Theorem 3 (see [18]). The operator is bounded on if and only if the sequence is bounded, where .

By Theorem 3, we can obtain that the operator is unbounded on Fock space , and it is bounded on .

Throughout this paper, we suppose that the sequence is bounded.

3. Hyers-Ulam Stability of Differentiation Operator

Let be normed spaces and consider a mapping . The following definition can be found in [14].

Definition 4. We say that has the Hyers-Ulam stability property (briefly, is HU-stable) if there exists a constant such that, for any , , and with , there exists an with and . The number is called a Hyers-Ulam stability constant (briefly HUS-constant) and the infimum of all HUS constants of is denoted by ; generally, is not a HUS constant of (see [9, 10]).

Theorem 5. Let be the differentiation operator on the Hilbert spaces of entire functions . Then the following statements are equivalent:(a)has Hyers-Ulam stability on ;(b)the sequence is bounded, where .

Proof. (b)⇒(a). Suppose that the sequence is bounded, and let . Since the polynomials are dense in , we just need to show that has Hyers-Ulam stability on the polynomials dense subspace. For each , we take any two polynomials and that satisfy , and can be represented by the orthonormal basis. Then if , , we have For any nonnegative integers and such that , we can get Hence,
Let be the function defined by
It is easy to check that , also; from (8), we obtain Next, assume that . It follows that Hence, Let Then ; by (12), we have Finally, if , we get Hence, Thus, from (16), it follows that Therefore, (a) holds.
(a)⇒(b). Suppose that is stable with Hyers-Ulam stability constant . For any nonnegative integer , let . Then , so there exists such that and . Hence, and consequently for any nonnegative integer . This completes the proof.

Example 6. We consider the comparison function ; that is, ; by a simple calculation, we have and (), and hence operator has Hyers-Ulam stability on the .

Example 7. If we put comparison function , that is, , we have and (), is unbounded, and hence operator is not Hyers-Ulam stable on the .

Example 8. We consider the comparison function ; we get , , (), and by a simple calculation, we have , , () and , , (), where is bounded; hence, operator has Hyers-Ulam stability on the .

Remark 9. From Theorem 5 and Examples 6–8, we can see that the Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions depends on the comparison functions . It shows that the comparison functions affects the behaviors of the operators and the functions in the corresponding Hilbert space .

Next, we will show that the best constant of Hyers-Ulam stability exists.

Theorem 10. If differentiation operator has the Hyers-Ulam stability on Hilbert spaces of entire functions , then and is a HUS constant of .

Proof. Suppose that has Hyers-Ulam stability on . By the proof of Theorem 5, we know that is a constant of the Hyers-Ulam stability of differentiation operator . Next, we show that it is the infimum of all the Hyers-Ulam stability constants. Let be an arbitrary Hyers-Ulam stability constant for ; put , and for any nonnegative integer , we can obtain , so there exists , such that and , and hence and so .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the editor and reviewers for their valuable comments and suggestions on the paper. This work is supported by the National Natural Science Foundation of China (Grant no. 11171022).