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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.

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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, ZBL06695787.

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.

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