Functional equations onA-orthogonal vectors

Abstract

LetH be a real (complex) Hilbert space with dimH ⩾ 3,A:H →H a continuous selfadjoint operator with dimA(H) > 2; we introduce onH a suitableA-orthogonality relation and study, in the class of the real (complex) functionals defined onH, two conditional functional equations — the Cauchy and the quadratic one — on the restricted domain of theA-orthogonal vectors. In this paper we determine the general solutions of these equations by theorems in which we establish the equivalence between each equation postulated on the whole space and the respective conditional equation. Our investigations have been motivated by incomplete studies on these conditional functional equations made in 1986 and 1966 by H. Drljevie and F. Vajzovie, respectively.