On $L^p(\mathbb{R}^+)$, we consider the following operator:
$$
Af:= f'',\qquad D(A):=\{u\in W^{2,p}(\mathbb{R}^+),u'(0)=0\}
$$
Now I want to know if this operator is a generator of a positive analytic semigroup of contractions. Can anybody give me a hint or references about this question. 



Positive analytic semigroup
You can check directly that it $A$ dispersive and that the numerical range is on a half-line, i.e.,
$$\langle Af,f^+\rangle \leq 0$$
holds for all $f\in D(A)$. You have to use integration by part and the fact that $W^{1,2}$ is a lattice. See for example Chapter 11.3 in 


*Bátkai, András; Kramar Fijavž, Marjeta; Rhandi, Abdelaziz*, [**Positive operator semigroups. From finite to infinite dimensions**](http://dx.doi.org/10.1007/978-3-319-42813-0), [ZBL06695787](https://zbmath.org/?q=an:06695787).


You only have to check then that the resolvent is nonempty, but this is just solving a simple ode.


Actually, since you can write down the resolvent directly for this operator, you can check the Hille-Yosida estimates directly as well, but it is more work.


