Improved polynomial decay for unbounded semigroups

We obtain polynomial decay rates for $C_0$-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigroups we improve upon previous results by, for example, removing a logarithmic loss on non-Hilbertian Banach spaces.