On a certain infinite-dimensional analogue of a theorem of Siegel (Q1080163)

Let \(W_ m\) be a Sobolev space of \(2\pi\)-periodic complex valued functions, \(A:W_ m\to W_ m\) a linear unitary operator and \(\phi:W_ m\to W_ m\) a Fréchet analytic nonlinear operator. The solvability of equation \[ (1)\quad H^{-1}\cdot (A+\phi)\cdot H(v)=Av \] with respect to H in some neighbourhood S of zero in \(W_ m\), where \(H:S\to W_ m\) is an analytic Fréchet diffeomorphism onto its image with a linear part identical in zero has been investigated in the paper. The author remarks that it is impossible to solve equation (1) for an arbitrary Fréchet analytic non-linearity \(\phi\). A class of non-linearities \(\phi\) for which equation (1) is solvable is pointed out.