Assuming that a Nash type inequality is satisfied by a non-negative self-adjoint operator
A
A
, we prove a Nash type inequality for the fractional powers
A
α
A^{\alpha }
of
A
A
. Under some assumptions, we give ultracontractivity bounds for the semigroup
(
T
t
,
α
)
(T_{t,{\alpha }})
generated by
−
A
α
-A^{\alpha }
.