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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 \(0\neq x\in H\). 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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