Remarks on ℓ₁ and ℓ∞ -maximal regularity for power-bounded operators

We discuss ℓ_p-maximal regularity of power-bounded operators and relate the discrete to the continuous time problem for analytic semigroups. We give a complete characterization of operators with ℓ₁ and ℓ∞-maximal regularity. We also introduce an unconditional form of Ritt's condition for power-bounded operators, which plays the role of the existence of an H^∞-calculus, and give a complete characterization of this condition in the case of Banach spaces which are L ₁-spaces, C(K)-spaces or Hilbert spaces.