Fredholm alternative for an elliptic Dirichlet problem

For a uniformly elliptic operator $L$ on a bounded regular domain, exactly one of the following equivalent descriptions applies to its homogeneous Dirichlet problem. If $\ker L^{\ast }=0$, every forcing has a unique solution. Otherwise $Lu=f$ is solvable exactly when $f$ is orthogonal to $\ker L^{\ast }$, and any two solutions differ by an element of $\ker L$. Moreover $\dim \ker L=\dim \ker L^{\ast }$.