Approximate orthogonality preserving property in Hilbert $C^*$-modules

In this talk, we define the concept of $(\delta, \varepsilon)$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta, \varepsilon)$-orthogonality preserving. In particular, if $\mathcal{E}$ is a full Hilbert $\mathcal{A}$-module with $\mathbb{K}(\mathcal{H})\subseteq \mathcal{A} \subseteq \mathbb{B}(\mathcal{H})$ and $T, S:\mathcal{E}\longrightarrow \mathcal{E}$ are two $\mathcal{A}$-linear mappings satisfying $\big|\langle Sx, Sy\rangle\big| = \|S\|^2\,|\langle x, y\rangle|$ for all $x, y\in \mathcal{E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta, \varepsilon)$-orthogonality preserving mapping and $$\Big\|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle\Big\|\leq \frac{4(\varepsilon - \delta)}{(1 - \delta)(1 + \varepsilon)} \|Tx\|\,\|Ty\|\qquad (x, y\in \mathcal{E}).$$