A. Sahleh - Department of Mathematic, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.
F. Olyani Nezhad - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

Let \mathcal{A} be a standard C^{*} -algebra. In this paper, we will study the continuity of \varepsilon -orthogonality preserving mappings between Hilbert \mathcal{A} -modules. Moreover, we will show that a local mapping between Hilbert \mathcal A -modules is   \mathcal A -linear. Furthermore, we will prove that for a pair of nonzero \mathcal A -linear mappings T,S:E\longrightarrow F , between Hilbert   \mathcal A -modules, satisfying   \varepsilon -orthogonality preserving property, there exists \gamma \in \mathbb{C} ,\begin{align*}    \Vert \langle T(x), S(y) \rangle-\gamma \langle x, y \rangle\Vert \leq \varepsilon \Vert T\Vert \Vert S\Vert \Vert x\Vert \Vert y \Vert,  \quad x, y \in E.\end{align*}Our results generalize the known ones in the context of Hilbert spaces.

ε-orthogonality preserving mappings, local mappings, Hilbert C*-modules