Characterization of orthogonally additive operators on sequence spaces (Q1097466)

Let X be a real sequence space which is AK as well as FK and with the property that for all \(t\in N\), \((x_ k)\in X:\| (x_ 1,x_ 2,...,x_ t,0,0,...)\| \leq \| (x_ k)\|\), where \(\|.\|\) denotes the paranorm on X. Let, moreover Y be a paranormed space. A mapping \(F: X\to Y\) is said to be orthogonally additive if \(F(x+y)=F(x)+F(y)\) whenever \(x_ ky_ k=0\) for all \(k\in N\), \(x=(x_ k)\), \(y=(y_ k).\) The author characterizes the continuous orthogonally additive maps from X to Y as follows. Theorem: \(F: X\to Y\) is orthogonally additive \(iff:\) F(x)\(=\sum_{k}g(k,x_ k)\) for \(X=(x_ d)\in X\) with \(g(k,x_ k): N\times R\to Y\) such that: i) \(g(k,0)=0\) for all \(k\in N\) ii) g(k,.) is continuous on R, for all k iii) \(P_ g: X\to cs(Y)\), where \(P_ g(x)=(g(k,x_ k))_ k\) and \(cs(Y)=\{(y_ n)\); \(y_ n\in Y\), for all n, and \(\sum_{n}y_ n\in Y\}.\)