Sternberg's theorem for Banach spaces (Q1058112)

The theorem of Sternberg and Chen concerning the equivalence of \(C^{\infty}\)-mappings in the neighbourhood of a fixed hyperbolic point is generalized to the infinite dimensional situation. Let \(E\) be a Banach space with the following property: There exists a representation \(V: E\to E\) of the germ of the identical mapping with all bounded derivatives \(\| V^{(k)}(x)\| =c_ k\) \((k=0,1,...,x\in E)\). The germs \(f,g: (E,0)\to (E,0)\) are called formally conjugate if there exists an invertible formal mapping \(\Phi: (E,0)\to (E,0)\) with continuous homogeneous components, such that \({\hat\Phi}(\hat f)=\hat g({\hat \Phi})\), where \(f,g\) are Taylor series on the origin. The germ \(f(x)=\Lambda x+O(\| x\|^ 2)\) is called to be hyperbolic if the linear operator \(\Lambda =f'(0)\) does not have spectrum at zero and on the unit circle. The following theorem is proved: If two germs of hyperbolic mappings are formally conjugated, then they are conjugated (i.e. there exists such a substitution \(\Phi: (E,0)\to (E,0)\), which is locally \(C^{\infty}\) and \(\Phi \circ f=g\circ \Phi)\).