A Remark on Contraction Semigroups on Banach Spaces

Let X be a complex Banach space and let J:XX^* be a duality sectionon X (that is, x,J(x)=||J(x)||||x||=||J(x)||²)=||x||²). Forany unit vector x and any (C⁰) contraction semigroup T={e^tA:t0}, Goldstein proved that if X is a Hilbert space and |T(t)x,j(x)|1 as t, then x is an eigenvector of A corresponding toa purel imaginary eigenvalue. In this article, we prove thata similar result holds if X is a strictly convex complex Banachspace.