We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \cdots \lambda_n\ge$ with corresponding eigenvectors $\{\phi_n\}_{n=1}^\infty$ such that $\{\phi_n\}_{n=1}^\infty$ can be normalized to be an orthonormal basis of $L^2(\Omega)$.
Questions: if $A$ is not symmetric. Say $A=\Delta + a\partial_x$, where $a\in C(\bar\Omega)$. Can the eigenvectors of $A$ still form an orthonormal basis of $L^2(\Omega)$? My guess is if $(\lambda-A)^{-1}$ is normal, then we can apply the spectral theory for compact normal operators. But is $(\lambda-A)^{-1}$ normal for large $\lambda$? Thanks!!!
