# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 46C20 Spaces with indefinite inner product 47A67 Representation theory of linear operators
Full Text:
##### 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] Den Broek, P. M. Van: Twistor space, Minkowski space and the conformal group. Physica A 122, 587-592 (1983) · Zbl 0591.53069 [3] Den Broek, P. M. Van: Symmetry transformations in indefinite metric spaces: A generalization of Wigner’s theorem. Physica A 127, 599-612 (1984) · Zbl 0599.20072 [4] Den Broek, P. M. Van: 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) [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) · Zbl 0155.56802