Mappings preserving approximate orthogonality in Hilbert $C^*$-modules

Abstract

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. 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 $\mathscr{E}$ is a full Hilbert $\mathscr{A}$-module with $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T, S:\mathscr{E}\longrightarrow \mathscr{E}$ are two linear mappings satisfying $\big|\langle Sx, Sy\rangle\big| = \|S\|^2\,|\langle x, y\rangle|$ for all $x, y\in \mathscr{E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta, \varepsilon)$-orthogonality preserving mapping. We also prove whenever $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T: \mathscr{E} \longrightarrow \mathscr{F}$ is a nonzero $\mathscr{A}$-linear $(\delta, \varepsilon)$-orthogonality preserving mapping between $\mathscr{A}$-modules, then $$\big\|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle\big\|\leq \frac{4(\varepsilon - \delta)}{(1 - \delta)(1 + \varepsilon)} \|Tx\|\,\|Ty\|\qquad (x, y\in \mathscr{E}).$$ As a result, we present some characterizations of the orthogonality preserving mappings.