Realization of operators and semigroups

Question

I don't understand something: it is said " we want to study the realization A of $\textit{A}$ in $L^2(\Omega)$ $\ldots$" speaking of operators and semigroups, in this case we are considering second order elliptic operators of the form $\textit{A}u = - \sum_{i,j=1}^{d}D_ja_{ij}D_iu + \sum_{i=1}^{d}b_iD_iu - \sum_{i=1}^{d}D_i(c_iu) + c_0u$ . Can anybody explain me what the realization means in this case? Thank you.