Natural curvature operators of bounded spectrum

Let M be a pseudo-Riemannian manifold of signature (p,q) where p?1 and q?1. If the Jacobi operator has pointwise bounded spectrum on the pseudo-sphere bundles of unit spacelike or timelike vectors, then M is pointwise Osserman. Similar results are established for other natural operators of Riemannian geometry. Rigidity phenomena in Lorentzian geometry are discussed.