Improving the Hartman-Grobman’s theorem, known result has proved that for a \(C^1\) hyperbolic system such that the derivative of the nonlinear perturbation is \(\alpha \) -Hölder continuous at the origin O, the differentiable linearization at O can be realized on a Banach space under a spectral bandwidth condition. In this paper, we improve this result to show that the differentiable linearization can be realized under a weaker spectral bandwidth condition, which is shown to be (almost) sharp by a counter example. In our proof, instead of decoupling the system by the stable and unstable foliations, we thoroughly decouple the system by the invariant foliation corresponding to each spectral subinterval of the linear part. Moreover, in order to overcome the difficulty that spectral gaps are not wide enough, we introduce a weaker smoothness than “ \(C^1\) plus \(\alpha \) -Hölder continuity of the derivative at O" for the decoupled system, which is enough to guarantee its differentiable linearization. As an effective application, we apply our result to abstract damped wave equations.
