  We consider an abstract non-negative self-adjoint operator $H$ on an
$L^2$-space. We derive a characterization for the restriction estimate $\|
dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} -
\frac{1}{p'}) -1}$ in terms of higher order derivatives of the semigroup
$e^{-tH}$. We provide an alternative proof of a result in [1] which asserts
that dispersive estimates imply restriction estimates. We also prove
$L^p-L^{p'}$ estimates for the derivatives of the spectral resolution of $H$.
