On the Resolvent of a Non-Self-Adjoint Differential Operator in Hilbert Spaces

The importance of studying non-self-adjoint differential operators is becoming more and more obvious to scientists. The non-self-adjoint operators appear in many branches of science. Today, these operators have many applications in kinetic theory and quantum mechanics to linearization of equations of mathematical physics. The spectrum of these operators is unstable and their resolvent is very unpredictable. In these operators, there is no general spectral theory and this causes problems in the study of these operators. In this paper, we consider a non-self-adjoint elliptic differential operator and study its resolvent.