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Sesquilinear-orthogonally quadratic mappings. (English) Zbl 0723.39009

Let X be a vector space over \(\Phi\) where \(\Phi\) is a field with characteristic \(\neq 2,3\) or 5. Assume that dim \(X\geq 3\). Let \((Y,+)\) be a 6-torsion-free abelian group, i.e., the multiplication by 6 in Y is injective. Let \(\phi: X\times X\to \Phi\) be a sesquilinear functional with respect to an automorphism \({\bar \phi}: \Phi \to \Phi,\) i.e., \(\phi\) is biadditive and \(\phi (\alpha x,\beta y)=\alpha {\bar \beta}\phi (x,y)\) for all \(x,y\in X\) and \(\alpha,\beta \in \Phi.\) \(x,y\in X\) are said to be \(\phi\)-orthogonal if \(\phi (x,y)=0.\) The author shows that if \(G: X\to Y\) satisfies the functional equation \(G(x+y)+G(x-y)=2G(x)+2G(y)\) for all \(x,y\in X\) with \(x\perp^{\phi}y\) then \(G(x+y)+G(x- y)=2G(x)+2G(y)\) for all \(x,y\in X\) provided there exists \(z\in X\) such that \(\phi (z,z)\neq 0\).

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
46C15 Characterizations of Hilbert spaces
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References:

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