Additive mappings preserving orthogonality between complex inner product spaces

TL;DR

This paper characterizes additive mappings between complex inner product spaces that preserve orthogonality, showing they are essentially scalar multiples of isometries or conjugate-isometries, extending real space results to complex spaces.

Contribution

It extends the characterization of orthogonality-preserving additive maps to complex inner product spaces, including conjugate-linear cases, generalizing previous real space theorems.

Findings

Additive orthogonality-preserving maps are scalar multiples of isometries or conjugate-isometries.

Equivalent conditions include orthogonality preservation and specific inner product relations.

Results generalize real space theorems to complex inner product spaces.

Abstract

Let $H$ and $K$ be two complex inner product spaces with dim$(X)\geq 2$. We prove that for each non-zero additive mapping $A:H \to K$ with dense image the following statements are equivalent: $(a)$ $A$ is (complex) linear or conjugate-linear mapping and there exists $\gamma >0$ such that $\| A (x) \| = \gamma \|x\|$, for all $x\in X$, that is, $A$ is a positive scalar multiple of a linear or a conjugate-linear isometry; $(b)$ There exists $\gamma_1 >0$ such that one of the next properties holds for all $x,y \in H$: $(b.1)$ $\langle A(x) |A(y)\rangle = \gamma_1 \langle x|y\rangle,$ $(b.2)$ $\langle A(x) |A(y)\rangle = \gamma_1 \langle y|x \rangle;$ $(c)$ $A$ is linear or conjugate-linear and preserves orthogonality in both directions; $(d)$ $A$ is linear or conjugate-linear and preserves orthogonality; $(e)$ $A$ is additive and preserves orthogonality in both directions; $(f)$ $A$ is additive and preserves orthogonality. This extends to the complex setting a recent generalization of the Koldobsky--Blanco--Turn\v{s}ek theorem obtained by W\'ojcik for real normed spaces.