Let $ A(x)=(a\_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum\_{i,j=1}^da\_{ij}(x)\xi\_i\xi\_j\geq\mu|\xi|^2 
\end{align}  

with $ \mu>0 $ being a positive constant and $ \xi\in\mathbb{R}^d $. Since the spectrum of $ L=-\operatorname{div}(A(x)\nabla) $, for $ \zeta\in\mathbb{C}\backslash\mathbb{R}\_+ $, $ (L-\zeta I)^{-1} $ exists. I read some paper and know that for some special $ A $ and $ p $, we can derive that
\begin{align}
\left\|(L-\zeta I)^{-1}\right\|\_{L^p\to L^p}\leq \frac{C}{1+|\zeta|}.
\end{align}
where $ C $ may depends on $ \arg\zeta $. I want to know what is the motivation to study such estimates and where can I use such estimates.



Applications and motivations of resolvent for elliptic operator
To begin with, the ellipticity condition is useless if you don't ask also that
$$\sum\_{i,j}a\_{ij}\xi\_i\xi\_j\le M|\xi|^2$$
for some finite constant $M$.


Now the resolvant estimate is used to define an operator $e^{-zL}$ for $\Re z>0$. In particular $S\_t=e^{-tL}$ is the semi-groups associated with the evolution equation
$$\partial\_tu+Lu=f$$
where $f(t,x)\in L^1(0,T;L^p)$ is given, as well as an initial value $u\_0\in L^p$.


