Stability of Derivations on Hilbert $C^*$-Modules

Abstract

Consider the functional equation ${\mathcal E}_1(f) = {\mathcal E}_2(f) ({\mathcal E})$ in a certain framework. We say a function $f_0$ is an approximate solution of $({\mathcal E})$ if ${\mathcal E}_1(f_0)$ and ${\mathcal E}_2(f_0)$ are close in some sense. The stability problem is whether or not there is an exact solution of $({\mathcal E})$ near $f_0$. In this paper, the stability of derivations on Hilbert $C^*$-modules is investigated in the spirit of Hyers--Ulam--Rassias.