Functional calculi, regularized semigroups and integrated semigroups

We characterize closed linear operators A, on a Banach space, for which the corresponding abstract Cauchy problem has a unique polynomially bounded solution for all initial data in the domain of $A^n$, for some nonnegative integer n, in terms of functional calculi, regularized semigroups, integrated semigroups and the growth of the resolvent in the right half-plane. We construct a semigroup analogue of a spectral distribution for such operators, and an extended functional calculus: When the abstract Cauchy problem has a unique $O(1 + t^k)$ solution for all initial data in the domain of $A^n$, for some nonnegative integer n, then a closed operator f(A) is defined whenever f is the Laplace transform of a derivative of any order, in the sense of distributions, of a function F such that $t → (1 + t^k)F(t)$ is in $L^{1}([0,∞))$. This includes fractional powers. In general, A is neither bounded nor densely defined