Spectral localization, power boundedness and invariant subspaces under Ritt's type condition

For a bounded linear operator T in a Banach space the Ritt resolvent condit ion parallel to R-lambda(T)parallel to less than or equal to C/\lambda - 1\ (\lambda\ > 1) Can be extended (changing the constant C) to any sector \ar g(lambda - 1)\ less than or equal to pi - delta, arccos(C-1) < delta < pi/2 . This implies the power boundedness of the operator T. A key result is tha t the spectrum sigma(T) is contained in a special convex closed domain. A g eneralized Ritt condition leads to a similar localization result and then t o a theorem on invariant subspaces.