  Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf
R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square
potential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ be
the operator norm of $\nabla^\alpha e^{-tH}$ from the Lorentz space
$L^{p,\sigma}({\bf R}^N)$ to $L^{q,\theta}({\bf R}^N)$, where
$\alpha\in\{0,1,2,\dots\}$. We establish both of upper and lower decay
estimates of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ and
study sharp decay estimates of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to
L^{q,\theta})}$. Furthermore, we characterize the Laplace operator $-\Delta$
from the view point of the decay of $\|\nabla^\alpha
e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$.
