Investigation of the Spectral Properties of a Non-Self-Adjoint Elliptic Differential Operator

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 (Au)(x) acting on Hilbert space and first investigate the spectral properties of space $H_1=L^2{\left(\mathrm{\Omega }\right)}^1$ . Then, as the application of this new result, the resolvent of the considered operator in $\ell$ -dimensional space Hilbert $H_{\ell }=L^2{\left(\mathrm{\Omega }\right)}^{\ell }$ is obtained utilizing some analytic techniques and diagonalizable way.