Einstein manifolds and curvature operator of the second kind

We prove that a compact Einstein manifold of dimension $n\geq 4$ with nonnegative curvature operator of the second kind is a constant curvature space by Bochner technique. Moreover, we obtain that compact Einstein manifolds of dimension $n\geq 11$ with $\left [ \frac{n+2}{4} \right ]$-nonnegative curvature operator of the second kind, $4\ (\mbox{resp.},8,9,10)$-dimensional compact Einstein manifolds with $2$-nonnegative curvature of the second kind and $5$-dimensional compact Einstein manifolds with $3$-nonnegative curvature of the second kind are constant curvature spaces. Combing with Li's result [10], we have that a compact Einstein manifold of dimension $n\geq 4$ with $\max\{4,\left [ \frac{n+2}{4} \right ]\}$-nonnegative curvature operator of the second kind is a constant curvature space.