Boundary conditions partial differential equations via theory of semigroups

Question

Consider a Cauchy problem in a Banach space with operator $\mathcal{A}$. This is obtained for example by starting from an evolution PDE in $\Omega \subset \mathbb{R}^n$ with operator $\mathcal{A}$ endowed some boundary conditions for example Dirichlet. I don't understand how essentially the boundary conditions "disappear" once the PDE is written in the abstract Cauchy problem. Usually the BC can be found in the domain of the operator $\mathcal{A}$ for example $D(\mathcal{A})={H^2(\Omega)} \cap H^1_0(\Omega)$ for the heat equation but I don't understand why we need to put them here... what does the domain where the operator is defined (e.g. $x \in X$ such that exists the limit $(T(h)-I)x/h \to 0 $ exists where T is the semigroup generated by $\mathcal{A}$) have to do with the boundary conditions? I mean the operator for the heat equation could be defined on all $H^2(\Omega)$ right?