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- Title
A functional equation related to Wigner's theorem.
- Authors
Huang, Xujian; Zhang, Liming; Wang, Shuming
- Abstract
An open problem posed by G. Maksa and Z. Páles is to find the general solution of the functional equation { ‖ f (x) - β f (y) ‖ : β ∈ T n } = { ‖ x - β y ‖ : β ∈ T n } (x , y ∈ H) where f : H → K is between two complex normed spaces and T n : = { e i 2 k π n : k = 1 , ⋯ , n } is the set of the nth roots of unity. With the aid of the celebrated Wigner's unitary-antiunitary theorem, we show that if n ≥ 3 and H and K are complex inner product spaces, then f satisfies the above equation if and only if there exists a phase function σ : H → T n such that σ · f is a linear or anti-linear isometry. Moreover, if the solution f is continuous, then f is a linear or anti-linear isometry.
- Subjects
FUNCTIONAL equations; QUADRATIC equations; INNER product spaces; NORMED rings
- Publication
Aequationes Mathematicae, 2024, Vol 98, Issue 3, p885
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-024-01042-8