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 \(c\in\mathbb{R}\).


46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
47A67 Representation theory of linear operators
Full Text: DOI arXiv


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.