Angle Preserving Mappings

Abstract

In this paper, we give some characterizations of orthogonality preserving mappings between inner product spaces. Furthermore, we study the linear mappings that preserve some angles. One of our main results states that if $\mathcal{X},\mathcal{Y}$ are real inner product spaces and $\theta \in (0,\pi )$, then an injective nonzero linear mapping $T:\mathcal{X}⟶\mathcal{Y}$ is a similarity whenever (i) $x\underset{\theta }{\angle }y\iff Tx\underset{\theta }{\angle }Ty$ for all $x,y\in \mathcal{X}$; (ii) for all $x,y\in \mathcal{X}$, $\Vert x\Vert =\Vert y\Vert$ and $x\underset{\theta }{\angle }y$ ensure that $\Vert Tx\Vert =\Vert Ty\Vert$. We also investigate orthogonality preserving mappings in the setting of inner product $C^{\ast }$-modules. Another result shows that if $\mathbb{K}(\mathcal{H})\subseteq \mathcal{A}\subseteq \mathbb{B}(\mathcal{H})$ is a $C^{\ast }$-algebra and $T:\mathcal{E}⟶\mathcal{F}$ is an $\mathcal{A}$-linear mapping between inner product $\mathcal{A}$-modules, then $T$ is orthogonality preserving if and only if $|x|\le |y|\Rightarrow |Tx|\le |Ty|$ for all $x,y\in \mathcal{E}$.