A Ritt-Kreiss condition: spectral localization and norm estimates

A new condition is introduced by generalizing the Ritt and Kreiss operators named $(\alpha ,\beta )$-RK condition. Geometrical properties of the spectrum for the case $\beta <1$ are studied, moreover it is shown that in that case 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 and interpolation result over Ritt and Kreiss operator on $L^p$ spaces.