### A functional calculus description of real interpolation spaces for sectorial operators

For a holomorphic function ψ defined on a sector we give a condition implying the identity ${(X,\left({A}^{\alpha}\right))}_{\theta ,p}=x\in X|{t}^{-\theta Re\alpha}\psi \left(tA\right)\in L{\u204e}^{p}((0,\infty );X)$ where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.