  The purpose of this article is to establish new lower bounds for the sums of
powers of eigenvalues of the Dirichlet fractional Laplacian operator
$(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain
$\Omega\subset{\mathbb R}^d$ with $d=2,$ $1\leq \alpha\leq 2$ and $d\geq 3,$
$0< \alpha\le 2$. Our main result yields a sharper lower bound, in the sense of
Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous
result in [36]. Furthermore, we give a result improving the bounds for
analogous elliptic operators in [19].
