The curvature operator on $\Lambda^2(TM)$ is defined on decomposable bivectors by $$g(\mathfrak{R}(X \wedge Y), V \wedge W) = R(X,Y,W,V)$$ and then extended by linearity to all elements of $\Lambda^2(TM)$. It is self-adjoint, so defines a symmetric bilinear form on $\Lambda^2(TM)$. If this form is positive definite, then all sectional curvatures are positive.


My question is: are there Riemannian manifolds with positive sectional curvature but with an indefinite curvature operator?


Positive sectional curvature means that $g(\mathfrak{R}(\alpha), \alpha) > 0$ for all *decomposable* bivectors, which does not seem to exclude the existence of an *indecomposable* bivector $\beta$ for which $g(\mathfrak{R}(\beta), \beta) < 0$. In dimension 3 all bivectors are decomposable, so counterexamples can exist only starting from dimension 4.


A second question: If the answer to the previous question is "yes", what pinching of the sectional curvature does imply the positivity of the curvature operator?



Positive sectional curvature does not imply positive definite curvature operator?
For the first question: positive curvature operator on a compact manifold implies that the manifold is diffeomorphic to a space form, i.e., a manifold of sectional curvature one. This is due to C. Boehm and B. Wilking [Manifolds with positive curvature operators are space forms](http://annals.math.princeton.edu/wp-content/uploads/annals-v167-n3-p10.pdf) Annals of Mathematics, 167 (2008), 1079–1097. 


Thus most known positively curved manifolds do not admit metrics with positive curvature operator. The simplest example is $CP^n$, $n>1$.


I do not know know much about how sectional curvature pinching can imply positive curvature operator but for example, you could trace references from J.-P. Bourguignon, H. Karcher,  [Curvature operators: pinching estimates and geometric examples](http://www.numdam.org/article/ASENS_1978_4_11_1_71_0.pdf) in Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 1, 71–92 where some estimates on the eigenvalues of the curvature operator in terms of sectional curvature pinching can be found. 


