## 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 $$0\neq x\in H$$. 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 $$c\in\mathbb{R}$$.

### MSC:

 46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) 47A67 Representation theory of linear operators
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### References:

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