A folk theorem in the spectral theory of \(C_ 0\)-semigroups
If A is the infinitesimal generator of a \(C_ 0\)-semigroup T(t), a classical theorem of Hille and Phillips relates the point spectrum of A and that of T(\(\xi)\) for \(\xi >0\). Specifically, if \(\mu\) is in the point spectrum of T(\(\xi)\) and \(\mu\) \(\neq 0\), then there exists \(\alpha_ 0\) in the point spectrum of A with \(\exp (\xi \alpha_ 0)=\mu\) and the null space of \(\mu\)-T(\(\xi)\) is the closed linear span of the null spaces of \(\alpha_ n-A\) for \(\alpha_ n=\alpha_ 0+2\pi in\xi^{-1}\) and n ranging over the integers. In this note we shall extend the Hille- Phillips theorem by proving that the null space of \((\mu -T(\xi))^ k\) is the closed linear span of the null spaces of \((\alpha_ n-A)^ k\) 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(\(\xi)\), and as an example we shall give an application to the theory of positive (in the sense of cone-preserving) linear operators.
infinitesimal generator of a \(C_ 0\)- semigroup
scientific article; zbMATH DE number 3967203