  We are concerned with periodic problems for nonlinear evolution equations at
resonance of the form $\dot u(t) = - A u(t) + F (t,u(t))$, where a densely
defined linear operator $A\colon D(A)\to X$ on a Banach space $X$ is such that
$-A$ generates a compact $C_0$ semigroup and $F\colon [0,+\infty)\times X \to
X$ is a nonlinear perturbation. Imposing appropriate Landesman--Lazer type
conditions on the nonlinear term $F$, we prove a formula expressing the fixed
point index of the associated translation along trajectories operator, in the
terms of a time averaging of $F$ restricted to $\mathrm{Ker} \, A$. By the
formula, we show that the translation operator has a nonzero fixed point index
and, in consequence, we conclude that the equation admits a periodic solution.
