The aim of this article is twofold. First we establish sharper lower bounds for the sums of eigenvalues of
(
−
Δ
)
1
2
|
D
,
(-\Delta )^{\frac {1}{2}}|_{D},
the Klein-Gordon operator restricted to a bounded domain
D
⊂
R
d
,
D\subset {\mathbb R}^d,
than the bounds obtained in works by E. Harrell; S. Yıldırım Yolcu; and G. Wei, H. Sun, and L. Zeng. Then we study upper bounds for the sums of negative powers of the eigenvalues of
(
−
Δ
)
1
2
|
D
.
(-\Delta )^{\frac {1}{2}}|_{D}.