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Characterization of orthogonally additive operators on sequence spaces. (English) Zbl 0635.46008
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 $$\sum_{n}y_ n\in Y\}.$$

MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)