A remark of contraction semigroups on Banach spaces
نویسنده
چکیده
Let X be a complex Banach space and let J : X → X∗ be a duality section on X (i.e. 〈x, J(x)〉 = ‖J(x)‖‖x‖ = ‖J(x)‖2 = ‖x‖2). For any unit vector x and any (C0) contraction semigroup T = {e tA : t ≥ 0}, Goldstein proved that if X is a Hilbert space and if |〈T (t)x, J(x)〉| → 1 as t → ∞, then x is an eigenvector of A corresponding to a purely imaginary eigenvalue. In this article, we prove the similar result holds if X is a strictly convex complex Banach space. 1991 Mathematics Subject Classification: 47B03, 47B15.
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