Let $(M,g)$ be a Riemannian manifold. The curvature operator on $\Lambda^2(TM)$ is defined on decomposable bivectors by $$g(\mathfrak{R}(X \wedge Y), V \wedge W) = R(X,Y,W,V)$$
where $g$ is given by the formula $g(X\wedge Y,Z\wedge W):=g(X,Z)g(Y,W)-g(X,W)g(Y,Z)$ and is extended by linearity to all of $\Lambda^2(TM)$. I want to know more about curvature operator and important theorems and open questions about it. Is there some good reference (Book) on this topic? If the answer is 'NO' can you introduce some good paper?
Thanks a lot.
Take a look at Einstein Manifolds by Besse, chapter $1$, section $H$ (it might help to also read section $G$). In particular, corollary $1.129$ tells you about the relationship between the curvature operator and Einstein in dimension four.