Einstein manifolds and curvature operator of the second kind

We prove that a compact Einstein manifold of dimension $n\ge 4$ 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 $n\ge 11$ with $\left[\frac{n+2}{4}\right]$-nonnegative curvature operator of the second kind, $4(resp\mathrm{.},8,9,10)$-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 $n\ge 4$ with $\max \{4,\left[\frac{n+2}{4}\right]\}$-nonnegative curvature operator of the second kind is a constant curvature space.