If A is the infinitesimal
generator of a C0-semigroup T(t), a classical theorem of Hille and Phillips
relates the point spectrum of A and that of T(ξ) for ξ > 0. Specifically,
if μ is in the point spectrum of T(ξ) and μ≠0, then there exists α0 in the
point spectrum of A with exp(ξα0) = μ and the null space of μ − T(ξ) is the
closed linear span of the null spaces of αn− A for αn= α0+ 2πinξ−1 and n
ranging over the integers. In this note we shall extend the Hille-Phillips
theorem by proving that the null space of (μ − T(ξ))k is the closed linear
span of the null spaces of (αn− A)k as n ranges over the integers. Such a
result is useful in relating the order of poles of the resolvent of A and the
order of poles of the resolvent of T(ξ), and as an example we shall give an
application to the theory of positive (in the sense of cone-preserving) linear
operators.
