Indefinite orthogonality preserving additive maps

Abstract.

Let H, K be real or complex complete indefinite inner product spaces of infinite dimension and let $\Phi :{\mathcal B}(H) \to {\mathcal B}(K)$ be an additive bijection. We show that Φ preserves indefinite orthogonality, i.e., $T^{\dag} S = 0 \Rightarrow \Phi (T)^{\dag} \Phi (S) = 0$ and $TS^{\dag} = 0 \Rightarrow \Phi (T)\Phi (S)^{\dag} = 0$ for any $T,S \in \mathcal{B}(H),$ if and only if there exist linear or conjugate linear bounded invertible operators U : H → K and V : K → H, which satisfy that both $U^{{{\dag}}} U$ and $V^{{{\dag}}} V$ are real scalar multiples of the identity, such that $\Phi (T) = UTV$ for all $T \in \mathcal{B}(H),$ where, for $R \in \mathcal{B}(H,K),R^{{\dag}}$ stands for the conjugate of R with respect to the indefinite inner products. A similar result is also true in the finite dimensional case if Φ(I) is assumed to be invertible.