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:

 [1] Bracci, L.; Morchio, G.; Strocchi, F., Wigner’s theorem on symmetries in indefinite metric spaces, Commun. math. phys., 41, 289-299, (1975) · Zbl 0303.46022 [2] Van den Broek, P.M., Twistor space, Minkowski space and the conformal group, Physica A, 122, 587-592, (1983) · Zbl 0591.53069 [3] Van den Broek, P.M., Symmetry transformations in indefinite metric spaces: A generalization of Wigner’s theorem, Physica A, 127, 599-612, (1984) · Zbl 0599.20072 [4] Van den Broek, P.M., Group representations in indefinite metric spaces, J. math. phys., 25, 1205-1210, (1984) · Zbl 0549.20030 [5] Matvejchuk, M., Gleason’s theorem in W^{*}J-algebras in spaces with indefinite metric, Internat. J. theoret. phys., 38, 2065-2093, (1999) · Zbl 0935.46052 [6] Molnár, L., A generalization of Wigner’s unitary-antiunitary theorem to Hilbert modules, J. math. phys., 40, 5544-5554, (1999) · Zbl 0953.46030 [7] Molnár, L., Generalization of Wigner’s unitary-antiunitary theorem for indefinite inner product spaces, Commun. math. phys., 201, 785-791, (2000) · Zbl 0957.46016 [8] Molnár, L., Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles, Commun. math. phys., 217, 409-421, (2001) · Zbl 1026.81006 [9] Omladic, M.; Šemrl, P., Additive mappings preserving operators of rank one, Linear algebra appl., 182, 239-256, (1993) · Zbl 0803.47026 [10] Ovchinnikov, P.G., Automorphisms of the poset of skew projections, J. funct. anal., 115, 184-189, (1993) · Zbl 0806.46069 [11] Uhlhorn, U., Representation of symmetry transformations in quantum mechanics, Ark. fysik, 23, 307-340, (1963) · Zbl 0108.21805 [12] Varadarajan, V.S., Geometry of quantum theory, (1968), D Van Nostrand Company, Inc Princeton · Zbl 0155.56802
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