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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 η 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 xH. We write x ̲· η y=0 if ηx 0 ,y 0 =0 holds for every x 0 x ̲ and y 0 y ̲. Suppose that T:H ̲H ̲ is a bijective ray transformation with the property that Tx ̲· η Ty ̲=0 if and only if x ̲· η y ̲=0 holds for every x ̲,y ̲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, Tx ̲=Ux ̲ for every 0xH. 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:HH induces a symmetry transformation T on H ̲ if and only if ηUx,Uy=cηx,y, (x,yH) holds for some constant c.

MSC:
46C20Spaces with indefinite inner product
47A67Representation theory of linear operators