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 condition ${\displaystyle \parallel R_{\lambda }\left(T\right)\parallel \le \frac{C}{|\lambda -1|}}$ (|λ| > 1) can be extended (changing the constant C) to any sector |arg(λ - 1)| ≤ π - δ, ${\displaystyle \arccos \left(C^{-1}\right)<\delta <\frac{\pi }{2}}$. This implies the power boundedness of the operator T. A key result is that the spectrum σ(T) is contained in a special convex closed domain. A generalized Ritt condition leads to a similar localization result and then to a theorem on invariant subspaces.