Orthogonally Additive 2-homogeneous Polynomials on Some Spaces

Abstract

For each unital algebra X and vector space Y over the complex field, a mapping f :X→ Y is called orthogonally additive, if for every a,b in X with ab = 0 = ba, we have f(a + b) = f(a) + f(b) . A 2-homogeneous polynomial P from X to Y is a mapping P:X → Y for which there is a multilinear symmetric operator α:X×X → Y such that P(x) = α(x,x), for every x \in X. Homogeneous polynomials have been studied by many mathematicians, such as [1], [2] and [3]. In this talk, we investigate orthogonally additive 2-homogeneous polynomials on a generalized matrix algebra. We then examine our results for a triangular algebra and also for the full matrix algebras.