Extremal properties of contraction semigroups on hilbert and banach spaces
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.
Bulletin of the London Mathematical Society
Goldstein, J. (1993). Extremal properties of contraction semigroups on hilbert and banach spaces. Bulletin of the London Mathematical Society, 25 (4), 369-376. https://doi.org/10.1112/blms/25.4.369