Extremal properties of contraction semigroups on $c_o$

Abstract

For any complex Banach space $X$, let $J$ denote the duality mapping of $X$. For any unit vector $x$ in $X$ and any ($C_0$) contraction semigroup $(T_t{)}_{t>0}$ on $X$, Baillon and Guerre-Delabriere proved that if $X$ is a smooth reflexive Banach space and if there is $x^{\ast }\in J(x)$ such that $|⟨T(t)x,J(x)⟩|\to 1$ as $t\to \infty$, then there is a unit vector $y\in X$ which is an eigenvector of the generator $A$ of $(T_t{)}_{t>0}$ associated with a purely imaginary eigenvalue. They asked whether this result is still true if $X$ is replaced by $c_o$. In this article, we show the answer is negative.