A remark on contraction semigroups on banach spaces


Let X be a complex Banach space and let J:X→X#x002A; 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 (C0) contraction semigroup T={etA: 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. © 1995 Oxford University Press.

Publication Title

Bulletin of the London Mathematical Society