Extremal properties of contraction semigroups on hilbert and banach spaces

Abstract

Let/be a unit vector and T = (7/t) = e?A: T ^ 0) be a (Co) contraction semigroup generated by A on a complex Hilbert space X. If KT( / ) /,/>l -* • 1 as t -* oo, t h e n / i s an eigenvector of A corresponding to a purely imaginary eigenvalue. If one allows X to be a Banach space, the same situation can be considered by replacing < r ( 0 / / > by ( f>(T(t) f) where 0 is a unit vector in X* dual t o / If / (T/t)f)/ -»•1 as t -• oo, is / a n eigenvector of A? The answer is sometimes yes and sometimes no. © 1993 London Mathematical Society.

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Bulletin of the London Mathematical Society

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