  The resolvent Krylov subspace method builds approximations to operator
functions $f(A)$ times a vector $v$. For the semigroup and related operator
functions, this method is proved to possess the favorable property that the
convergence is automatically faster when the vector $v$ is smoother. The user
of the method does not need to know the presented theory and alterations of the
method are not necessary in order to adapt to the (possibly unknown) smoothness
of $v$. The findings are illustrated by numerical experiments.
