Nash type inequalities for fractional powers of non-negative self-adjoint operators

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