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 ${\displaystyle A^n}$, 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 ${\displaystyle O(1+t^k)}$ solution for all initial data in the domain of ${\displaystyle A^n}$, 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 ${\displaystyle t\to (1+t^k)F\left(t\right)}$ is in ${\displaystyle L^1\left([0,\infty )\right)}$. This includes fractional powers. In general, A is neither bounded nor densely defined.