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Dec 5, 2017 at 0:25 comment added Delio Mugnolo Good answer. Concerning 6., let me add that a good description of the domain of $\Delta_1$ is proved by Pazy in §7.3, Thm. 3.10 of his book on semigroups.
Sep 22, 2017 at 8:14 history answered Jochen Glueck CC BY-SA 3.0