We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space \({\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} \) as $$ {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}} $$ for a wide range of holomorphic functions ψ1, ψ2 and show that in this space the operator A has a bounded operator-valued H∞-functional calculus which is even R-bounded in case p < ∞. This generalizes results of Dore, Clement and Pruss. Consequences are a Da Prato-Grisvard theorem for injective commuting sectorial operators A, B and an adjoint-free proof of McIntosh’s theorem. Finally, we investigate the functional calculus properties for non-invertible operators A on the spaces \({\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} \).