Operators with resolvent of bounded characteristic

It is proved that if T is a bounded linear operator on a complex Banach space such that the sequence of norms {Tⁿ, n = 1,2, } is of power growth, then the resolvent of T is of bounded characteristic on the open unit disc D in the complex plane, if and only if T is annihilated by a nonzero holomorphic function on D with infinitely differentiable boundary values. This is analogous to the known result that an operator on a complex Banach space has rational resolvent, if and only if it is annihilated by a nonzero polynomial.