Branson's $Q$-curvature in Riemannian and Spin Geometry

On a closed $n$-dimensional manifold, $n\ge 5$, we compare the three basic conformally covariant operators: the Paneitz–Branson, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's $Q$-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's $Q$-curvature. Equality cases are also characterized.