zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 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.

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