Sesquilinear-orthogonally quadratic mappings
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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.
AMS (1980) subject classificationPrimary 39B70, 46C10
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- Aczél, J.,The general solutions of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases. Glas. Mat.-Fiz. Astronom.20 (1965), 65–73.Google Scholar
- Drljević, H.,On a functional which is quadratic on A-orthogonal vectors. Publ. Inst. Math. (Beograd) (N.S.)54 (1986), 63–71.Google Scholar
- Maksa, Gy., Szabó, Gy. andSzékelyhidi, L.,Equations arising from the theory of orthogonally additive and quadratic functions. C.R. Math. Rep. Acad. Sci. Canada10 (1988), 295–300.Google Scholar
- Rätz, J.,On orthogonally additive mappings. Aequationes Math.28 (1985), 35–49.Google Scholar
- Sundaresan, K., andKapoor, O. P.,T-orthogonality and nonlinear functionals on topological vector spaces. Canad. J. Math.25 (1973), 1121–1131.Google Scholar
- Szabó, Gy.,ϕ-orthogonally additive mappings, I. To appear in Acta Math. Hungar.Google Scholar
- Vajzović, F.,Über das Funktional H mit der Eigenschaft: (x,y) = 0 ⇒ H(x + y) + H(x − y) = 2H(x) + 2H(y). Clas. Mat. Ser. III2(22) (1967), 73–81.Google Scholar