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 H1 L2(Ω) 1. (en, as the application of this new result, the resolvent of the considered operator in l-dimensional space Hilbert Hl L2(Ω) l is obtained utilizing some analytic techniques and diagonalizable way.