Characterizations of domains of fractional powers for non-negative operators

It is known that a closed linear operator A, defined densely on a Banach space X, is non-negative if and only if −A generates a bounded analytic γ  -times resolvent family for some 0<γ<20<γ<2. In this paper, by using such resolvent families as well as the Komatsu representations, we characterize systematically the domains of fractional powers of non-negative operators on Banach spaces. Gaps between the subset ⋃α>0D(Aα)⋃α>0D(Aα) and the whole space X are discussed as well.