Mohamed S. ElBialy
Local Contractions of Banach Spaces
and Spectral Gap Conditions
(111K, LaTeX 2e)

ABSTRACT.       In this work we study the linearization problem for a $C^{k,1}, k\geq 1,$
         contraction of a Banach space $E$ near a fixed point which satisfies
         a spectral gap condition and a narrow band condition both of order $k$.
     We also assume that the part of the spectrum in each band
     satisfies a finite nonresonant condition of
        order $k$ relative to itself together  with the part that lies
        in the larger bands.
     We show that there is a $C^{k,\gb}$ linearization for sufficiently small
        $\gb > 0$.
     We give a precise estimate on $\gb$ in terms of the gap and
     band conditions.