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

G 1 , then there exists a homomorphism H : G 1 → G 2 with d(h(x),H(x)) < ε for all x ∈ G 1 ? In the following years, Hyers affirmatively answered the question of Ulam for the case where G 1 and G 2 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

In the following years, Hyers affirmatively answered the question of Ulam for the case where  1 and  2 are Banach spaces (see [2]).Furthermore, the result of Hyers has been generalized by Rassias (see [3]).
In this paper, we discuss the Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions  2 () 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.

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][21][22]).
Let us call an entire function () = ∑ ∞ =0     a comparison function if   > 0 for each , and the sequence of ratios  +1 /  decreases to zero as  increases to ∞.For each comparison function (), we define  2 () to be the Hilbert space of power series for which It is easy to check that each element of  2 () 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  2 () is given by and the functions form an orthonormal basis for  2 ().We can see that the polynomials are dense in  2 ().
Example 1.We consider the comparison function Throughout this paper, let  :  2 () →  2 () be the differentiation operator defined by An important result about  is the following theorem.
By Theorem 3, we can obtain that the operator  is unbounded on Fock space  2 ( 1 ), and it is bounded on  2 ( 2 ).
Throughout this paper, we suppose that the sequence {  } ∞ =1 is bounded.

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  > 0 such that, for any  ∈ (),  > 0, and  ∈  with ‖ − ‖ ≤ , there exists an  0 ∈  with  0 =  and ‖ −  0 ‖ ≤ .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]).