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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 $\underline x \cdot_\eta y=0$ if $\langle\eta x_0,y_0 \rangle=0$ 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 $T \underline x\cdot_\eta T\underline y=0$ if and only if $\underline x\cdot_\eta \underline y = 0$ 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 $\langle\eta Ux,Uy \rangle= c\langle \eta x,y\rangle$, $(x,y\in H)$ holds for some constant $c\in\bbfR$.

MSC:
46C20Spaces with indefinite inner product
47A67Representation theory of linear operators
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References:
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