Angle Preserving Mappings
Mohammad Sal Moslehian, Ali Zamani, Michael Frank
TL;DR
This work investigates how angle and orthogonality structures are preserved by mappings between inner product spaces and their extensions to inner product C*-modules. It proves that, in real inner product spaces, an injective nonzero linear map that preserves angle relations for all pairs must be a similarity, equivalently scaling norms uniformly and preserving inner products up to a positive scalar; it also provides a constructive criterion involving a unique scalar λ that realizes a prescribed angle under perturbations. The results extend to inner product C*-modules by showing that orthogonality preservation is equivalent to a scalar-compatible relation ⟨Tx, Ty⟩ = γ⟨x, y⟩ and a corresponding |Tx|–|x| order-preserving property, with additional insights for local maps and modules over C*-algebras containing K(H). Overall, the paper links angle- and orthogonality-preserving properties with similarity-type behavior across both Hilbert spaces and Hilbert C*-modules, offering a unified view of structure-preserving maps in these settings.
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 $θ\in(0, π)$, then an injective nonzero linear mapping $T:\mathcal{X}\longrightarrow \mathcal{Y}$ is a similarity whenever (i) $x\undersetθ{\angle} y\, \Leftrightarrow \,Tx\undersetθ{\angle} Ty$ for all $x, y\in \mathcal{X}$; (ii) for all $x, y\in \mathcal{X}$, $\|x\|=\|y\|$ and $x\undersetθ{\angle} y$ ensure that $\|Tx\|=\|Ty\|$. We also investigate orthogonality preserving mappings in the setting of inner product $C^{*}$-modules. Another result shows that if $\mathbb{K}(\mathscr{H})\subseteq\mathscr{A}\subseteq\mathbb{B}(\mathscr{H})$ is a $C^{*}$-algebra and $T\,:\mathscr{E}\longrightarrow \mathscr{F}$ is an $\mathscr{A}$-linear mapping between inner product $\mathscr{A}$-modules, then $T$ is orthogonality preserving if and only if $|x|\leq|y|\, \Rightarrow \,|Tx|\leq|Ty|$ for all $x, y\in \mathscr{E}$.