@misc{10481/106307,
year = {2025},
month = {8},
url = {https://hdl.handle.net/10481/106307},
abstract = {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) ≥ 2, and let
A ∶ H → 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 ∈ H we have A(iz) ∈ {±iA(z)}.
(c) There exists a non-zero point z ∈ H such that A(iz) ∈ {±iA(z)}.
(d) There exists a non-zero point z ∈ H such that iA(z) ∈ A(H).
The mapping A is neither complex-linear nor conjugate-linear if, and only if, there
exists a non-zero x ∈ H such that iA(x) ∉ A(H) (equivalently, for every non-zero
x ∈ H, iA(x) ∉ 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 fnite dimensional with dim(K) < 2dim(H).},
organization = {Universidad de Granada / CBUA},
publisher = {Springer Nature},
keywords = {Birkhof-orthogonality},
keywords = {Euclidean orthogonality},
keywords = {Orthogonality preserving additive mappings},
title = {An algebraic characterization of linearity for additive maps preserving orthogonality},
doi = {10.1007/s43034-025-00454-0},
author = {Li, Lei and Liu, Siyu and Peralta Pereira, Antonio Miguel},
}