Sesquilinearorthogonally quadratic mappings
 Gy. Szabó
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessSummary
The conditional Jordanvon 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 toAorthogonalities on a Hilbert space, which satisfyx ⊥^{ A } y ⇔〈Ax, y〉 = 0 whereA is a selfadjoint operator. In particular, Drljević in 1986 determined the continuous scalar valued solutions and recently Fochi showed that theAorthogonally 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 sesquilinearorthogonally 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 2dimensional counter example is presented. Finally, an interesting concept of orthogonality is suggested for possible future investigation.
 Aczél, J. (1965) 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: pp. 6573
 Drljević, H. (1986) On a functional which is quadratic on Aorthogonal vectors. Publ. Inst. Math. (Beograd) (N.S.) 54: pp. 6371
 Fochi, M. (1989) Functional equations on Aorthogonal vectors. Aequationes Math. 38: pp. 2840 CrossRef
 Maksa, Gy., Szabó, Gy., Székelyhidi, L. (1988) Equations arising from the theory of orthogonally additive and quadratic functions. C.R. Math. Rep. Acad. Sci. Canada 10: pp. 295300
 Rätz, J. (1985) On orthogonally additive mappings. Aequationes Math. 28: pp. 3549
 Sundaresan, K., Kapoor, O. P. (1973) Torthogonality and nonlinear functionals on topological vector spaces. Canad. J. Math. 25: pp. 11211131
 Szabó, Gy.,ϕorthogonally additive mappings, I. To appear in Acta Math. Hungar.
 Vajzović, F. (1967) Über das Funktional H mit der Eigenschaft: (x,y) = 0 ⇒ H(x + y) + H(x − y) = 2H(x) + 2H(y). Clas. Mat. Ser. III 2: pp. 7381
 Title
 Sesquilinearorthogonally quadratic mappings
 Journal

aequationes mathematicae
Volume 40, Issue 1 , pp 190200
 Cover Date
 19901201
 DOI
 10.1007/BF02112295
 Print ISSN
 00019054
 Online ISSN
 14208903
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Keywords

 Primary 39B70, 46C10
 Industry Sectors
 Authors

 Gy. Szabó ^{(1)}
 Author Affiliations

 1. Institute of Mathematics, L. Kossuth University, Pf. 12, H4010, Debrecen, Hungary