Investigation of the spectral properties of a non-self-adjoint elliptic differential operator (Q2038313)

Summary: Non-self-adjoint operators have many applications, including quantum and heat equations. On the other hand, the study of these types of operators is more difficult than that of self-adjoint operators. In this paper, our aim is to study the resolvent and the spectral properties of a class of non-self-adjoint differential operators. So we consider a special non-self-adjoint elliptic differential operator \((\mathrm{Au})(x)\) acting on Hilbert space and first investigate the spectral properties of space \(H_1= L^2 (\Omega)^1\). Then, as the application of this new result, the resolvent of the considered operator in \(\ell\)-dimensional space Hilbert \(H_\ell= L^2 (\Omega)^\ell\) is obtained utilizing some analytic techniques and diagonalizable way.