On uniqueness of fractional powers of multi-valued linear operators and the incomplete Cauchy problem

Abstract

We obtain uniqueness of additive families {A t } t>0 of fractional powers of a multi-valued sectorial linear operator A ≡ A 1 in a Banach space, satisfying a certain kind of continuity with respect to the exponent and a spectral property, from uniqueness of the solutions of the second-order incomplete Cauchy problem associated with A. We show the close relationship between the multiplicativity and the uniqueness of fractional powers.