Quasi-Distribution Semigroups and Integrated Semigroups

SupposeXis a Banach space. We introduce the concept of quasi-distribution semigroups onX, as a generalization of the concept of distribution semigroups introduced by Lions in [19]. In our approach, the generatorAof a quasi-distribution semigroup may not be densely defined. Also introduced is a functional calculus forAin terms of the Fourier transform. For fixedr>0,k∈N∪{0}, define order (r, k) for a quasi-distribution semigroup and define an Fr, kfunctional calculus. We prove thatAgenerates a (k+1)-times integrated semigroup of exponential typerwith local Lipschitz continuity if and only ifAgenerates a quasi-distribution semigroup of order (r, k) if and only ifAhas an Fr, kfunctional calculus. WhenAis densely defined, a quasi-distribution semigroup reduces to a distribution semigroup in the sense of Lions.