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Orthogonality preserving transformations on indefinite inner product spaces: Generalization of Uhlhorn’s version of Wigner’s theorem. (English) Zbl 1010.46023
The author obtains the following result as a corollary of the generalized Wigner-Uhlhorn theorem: Let $\eta$ be an invertible bounded linear operator on a Hilbert space $H$ with dimension not less than 3, and $x$ be the set of all nonzero scalar multiples of $x\in H$. We write $\underline{x}{·}_{\eta }y=0$ if $〈\eta {x}_{0},{y}_{0}〉=0$ holds for every ${x}_{0}\in \underline{x}$ and ${y}_{0}\in \underline{y}$. Suppose that $T:\underline{H}\to \underline{H}$ is a bijective ray transformation with the property that $T\underline{x}{·}_{\eta }T\underline{y}=0$ if and only if $\underline{x}{·}_{\eta }\underline{y}=0$ holds for every $\underline{x},\underline{y}\in \underline{H}$. That is, $T$ is a symmetry transformation. If $H$ is real, then $T$ is induced by an invertible bounded linear operator $U$ on $H$. That is, $T\underline{x}=\underline{Ux}$ for every $0\ne x\in H$. If $H$ is complex, then $T$ is induced by an invertible bounded linear or conjugate-linear operator $U$ on $H$. The operator $U$ inducing $T$ is unique up to muliplication by a scalar. That is, if $H$ is real, then the invertible bounded linear operator $U:H\to H$ induces a symmetry transformation $T$ on $\underline{H}$ if and only if $〈\eta Ux,Uy〉=c〈\eta x,y〉$, $\left(x,y\in H\right)$ holds for some constant $c\in ℝ$.

##### MSC:
 46C20 Spaces with indefinite inner product 47A67 Representation theory of linear operators