An optimal linear approximation for a class of nonlinear operators between uniform algebras
Abstract
Assume that A,B are uniform algebras on compact Hausdorff spaces X and Y, respectively. Let T:A→B be a map (nonlinear in general) satisfying T(A−1)=B−1 and T1=1. We show that, if there exist constants α,β≥1 such that β−1‖f⋅g−1‖≤‖Tf⋅(Tg)−1‖≤α‖f⋅g−1‖ for all f∈A and g∈A−1, then there exists a homeomorphism τ:∂B→∂A between the Šilov boundaries of A and B such that (αβ)−1|f(τ(y))|≤|(Tf)(y)|≤αβ|f(τ(y))| for all f∈A and for all y∈∂B. In particular ∂A and ∂B are homeomorphic. Moreover we give an example which shows that the multiple αβ in the above inequality is best possible.
Publication Title
Journal of Mathematical Analysis and Applications
Recommended Citation
Dong, Y., Lin, P., & Zheng, B. (2018). An optimal linear approximation for a class of nonlinear operators between uniform algebras. Journal of Mathematical Analysis and Applications, 467 (1), 432-445. https://doi.org/10.1016/j.jmaa.2018.07.016
