A Remark on Contraction Semigroups on Banach Spaces

Let X be a complex Banach space and let J : X → X * be a duality section on X (that is, 〈 x , J ( x )〉=∥ J ( x )∥∥ x ∥=∥ J ( x )∥ 2 )=∥ x ∥ 2 ). For any unit vector x and any ( C 0 ) contraction semigroup T ={e tA : t ⩾0}, 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 to a purel imaginary eigenvalue. In this article, we prove that a similar result holds if X is a strictly convex complex Banach space.