Gaussian estimates and regularized groups

We show that if a bounded analytic semigroup $\{T(z)\}_{ \operatorname{Re}z>0}$ on $L^2({\boldsymbol{\Omega}} )$ $({\boldsymbol{\Omega}} \subset\mathbf{R} ^n)$ satisfies a Gaussian estimate of order $m$ and $A_p$ is the generator of its consistent semigroup on $L^p({\boldsymbol{\Omega}} )$ $(1\le p<\infty)$, then $iA_p$ generates a $(1-A_p)^{-\alpha}$-regularized group on $L^p({\boldsymbol{\Omega}} )$ where $\alpha>2n |\frac{1}{2}-\frac{1}{p}|$. We obtain the estimate of $(\lambda-A_p)^{-1}$ ($|\operatorname{arg}\lambda|<\pi$) and the $p$-independence of $\sigma(A_p)$, and give applications to Schrödinger operators and elliptic operators of higher order.