Mappings preserving approximate orthogonality in Hilbert $C^{\ast }$-modules

Abstract

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^{\ast }$-modules. We define the concept of $(\delta ,\epsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta ,\epsilon )$-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 $T,S:\mathcal{E}⟶\mathcal{E}$ are two linear mappings satisfying $|⟨Sx,Sy⟩|=\Vert S{\Vert }^2|⟨x,y⟩|$ for all $x,y\in \mathcal{E}$ and $\Vert T-S\Vert \le \theta \Vert S\Vert$, then we show that $T$ is a $(\delta ,\epsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb{K}(\mathcal{H})\subseteq \mathcal{A}\subseteq \mathbb{B}(\mathcal{H})$ and $T:\mathcal{E}⟶\mathcal{F}$ is a nonzero $\mathcal{A}$-linear $(\delta ,\epsilon )$-orthogonality preserving mapping between $\mathcal{A}$-modules, then $$\Vert ⟨Tx,Ty⟩-\Vert T{\Vert }^2⟨x,y⟩\Vert \le \frac{4(\epsilon -\delta )}{(1-\delta )(1+\epsilon )}\Vert Tx\Vert \Vert Ty\Vert (x,y\in \mathcal{E}).$$ As a result, we present some characterizations of the orthogonality preserving mappings.

21 pages, to appear in Math. Scand