  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.
