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Perturbations of norm-additive maps between continuous function spaces
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 10 October 2024, pp. 1-16
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Let
$X, Y$ be two locally compact Hausdorff spaces and 
$T:C_0(X)\rightarrow C_0(Y)$ be a standard surjective ɛ-norm-additive map, i.e.
\begin{equation*}\big|\|T(f)+T(g)\|-\|f+g\|\big|\leq \varepsilon,\;{\rm for\;all}\; f, g\in C_0(X).\end{equation*}Then there exist a homeomorphism
$\varphi:Y\rightarrow X$ and a continuous function 
$\lambda:Y\rightarrow\lbrace\pm1\rbrace$ such that
\begin{equation*}|T(f)(y)-\lambda(y)f(\varphi(y))|\leq\frac{3}{2}\varepsilon,\;{\rm for\;all}\;y\in Y,\;f\in C_0(X).\end{equation*}The estimate ‘
$\frac{3}{2}\varepsilon$’ is optimal. And this result can be regarded as a new nonlinear extension of the Banach–Stone theorem.
