Sesquilinear-orthogonally quadratic mappings
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- Szabó, G. Aeq. Math. (1990) 40: 190. doi:10.1007/BF02112295
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The conditional Jordan-von Neumann functional equation for a mappingG: (X, +, ⊥) → (Y, +), that is,G(x + y) + G(x−y) = 2G(x) + 2G(y) for allx, y ∈ X withx ⊥ y, was first studied by Vajzović in 1966. He gave the general form of the continuous scalar valued solutions of (*) on a Hibert space with its natural orthogonality. Later his result was generalized toA-orthogo-nalities on a Hilbert space, which satisfyx ⊥Ay ⇔〈Ax, y〉 = 0 whereA is a selfadjoint operator. In particular, Drljević in 1986 determined the continuous scalar valued solutions and recently Fochi showed that theA-orthogonally quadratic functionals are exactly the quadratic ones.
Here we further generalize their results to a symmetric orthogonality induced by a sesquilinear form on a vector space and for arbitrary mappings with values in an abelian group. The main result states that such a mapping can satisfy (*) only if it is quadratic. In the proof extensive use is made of the theory of sesquilinear-orthogonally additive mappings as developed in an earlier paper of ours.
The above mentioned results are valid only for the cases of dimension ⩾ 3 and a 2-dimensional counter example is presented. Finally, an interesting concept of orthogonality is suggested for possible future investigation.