Hello. I cannot understand the conclusion of the study of the diffusion equation in "one-parameter semigroups of positive operators". More specifically the part where it is said that $(T(t))_{t\geq 0}$ is the semigroup generated by the closure of the second derivative with domain $D(B)$.
Question 1. Does this mean that $A=\overline{B}$, right?
Question 2. The above is true since; $E_0$ dense in $D(A)$ with $\left\|\cdot \right\|_{A}$ (graph norm of $A$) and $\left.A\right|_{E_0}=B \Rightarrow A = \overline{B}?$ (This is part of an overall result?)
I ask this to see if I understand it correctly.
I attach in images the definitions that appear in the book for clarity.
