An algebraic characterization of linearity for additive maps preserving orthogonality
Lei Li, Siyu Liu, Antonio M. Peralta
TL;DR
The paper addresses when an additive map $A:H\to K$ between complex inner product spaces that preserves Euclidean orthogonality must be complex-linear or conjugate-linear. It proves a precise algebraic characterization: such maps are either complex-linear or conjugate-linear precisely when, for all $z\in H$, $A(i z)\in\{\pm i A(z)\}$ (and equivalent conditions), with the non-linear case characterized by the existence of a nonzero $x$ with $i A(x)\notin A(H)$. It then shows that every nonzero orthogonality-preserving additive map is a positive scalar multiple of a real-linear isometry, and leverages pure/mixed complex-type analysis to obtain the equivalent criteria and obstructions. As corollaries, norm-dense range or certain finite-dimensional range constraints guarantee complex-linearity or conjugate-linearity, and a lifting to completions yields a representation $A=\gamma T$ where $T$ is a real-linear isometry whose complex-type behavior determines the overall structure. These results unify and extend known theorems on orthogonality preservers, clarifying when automatic linearity arises from preserving orthogonality alone.
Abstract
We study when an additive mapping preserving orthogonality between two complex inner product spaces is automatically complex-linear or conjugate-linear. Concretely, let $H$ and $K$ be complex inner product spaces with dim$(H)\geq 2$, and let $A: H\to K$ be an additive map preserving orthogonality. We obtain that $A$ is zero or a positive scalar multiple of a real-linear isometry from $H$ into $K$. We further prove that the following statements are equivalent: $(a)$ $A$ is complex-linear or conjugate-linear. $(b)$ For every $z\in H$ we have $A(i z) \in \{\pm i A(z)\}$. $(c)$ There exists a non-zero point $z\in H$ such that $A(i z) \in \{\pm i A(z)\}$. $(d)$ There exists a non-zero point $z\in H$ such that $i A(z) \in A(H)$. The mapping $A$ neither is complex-linear nor conjugate-linear if, and only if, there exists a non-zero $x\in H$ such that $i A(x)\notin A(H)$ (equivalently, for every non-zero $x\in H$, $i A(x)\notin A(H)$). Among the consequences we show that, under the hypothesis above, the mapping $A$ is automatically complex-linear or conjugate-linear if $A$ has dense range, or if $H$ and $K$ are finite dimensional with dim$(K)< 2\hbox{dim}(H)$.