A Ritt–Kreiss condition: Spectral localization and norm estimates

A new condition is introduced by generalizing the Ritt and Kreiss operators, named the $(\alpha ,\beta )$-RK condition. Geometrical properties of the spectrum for $\beta \lt 1$ are studied and moreover it is shown that then if $\alpha + \beta = 1$ the operator is Ritt. Estimates for the power and power differences norms for this type of operators are also studied. Lastly, we apply this theory to obtain an interpolation result for Ritt and Kreiss operators on $L^p$ spaces.