I'm studying about Semigroups in Parabolic equations this semester and I'm having a really hard time understanding how these complex line integrals behave from times to times (my complex analysis background contains is elementary). So, we have the following definition:
and we have shown that the integral is well-defined exploiting mostly the resolvent estimate for A being sectorial. Later in the lecture, when we started considering strong contunity, the following observation arose:
However this observation seems odd to me since with the same estimate we have shown the convergence of the integral in the above definition. I know in general that for $a>0$ the improper integral $\int_a^\infty \frac{1}{x^p}\;dx$ is convergent if $p>1$ which it seems is what is used here but I struggle relate it to this estimate.
I would appreciate if somebody could make the former computation in a detailed way in order to be clear why we can not just take the limit $t \to 0$ in $e^{-tA}x$ for all $x \in X$.
Thanks a lot in advance!
So, I think that I understood the relation. In order for $(e^{-tA})_{t\ge 0}$ to be a strongly continuous semigroup, the following limit must hold:
$\lim_\limits{t \to 0} e^{-tA}x=x, \quad \forall x\in X$
Now if we let $t \to 0$ we observe from the above definition that $\forall x\in X$, the integral that we will obtain is $\int_{\omega+\gamma^{r,\beta}} R(\lambda,-A)x\;d\lambda$ whose convergence can not be guaranteed by the resolvent estimate. We need some quadratic estimate in order for the integral to converge and hence we can not have it for all $x\in X$.