An extremal property of contraction semigroups in Banach spaces (Q1909534)

Let \(f\) be a nonzero vector and \(T= \{T(t)= e^{tB}: t\geq 0\}\) be a \(C_0\) contraction semigroup generated by \(B\) in a complex Banach space \(X\). If \(|\langle T(t) f, x^*\rangle|\to |\langle f, x^*\rangle|\) as \(t\to \infty\) for every \(x^*\) in the dual space \(X^*\), then \(f\) is an eigenvector of \(B\) corresponding to a purely imaginary eigenvalue. This seems to be the adequate generalization of the Hilbert space situation, where the assumption suffices that the single functional \(x^*\) is represented by \(f\) itself, whereas the corresponding single functional assumption in a general Banach space does not.