Spectral Properties of Non-Self-Adjoint Elliptic Differential Operators in Hilbert Space

Let Omega be a bounded domain in R-n with smooth boundary partial derivative Omega. In this article, we will investigate the spectral properties of a non-self adjoint elliptic differential operator (Au)(x) = - Sigma(n)(i,j=1) omega(2 alpha)(x)a(ij) (x)mu(x)u(xi)' (x))(xj)', acting in the Hilbert space H = L-2(Omega). with Dirichlet-type boundary conditions. Here a(ij)(x) = <(a(ij)(x))over bar> (i, j = 1, ..., n), a(ij)(x) is an element of C-2<((Omega))over bar>, and the functions a(ij)(x) satisfies the uniformly elliptic condition, and let 0 <= alpha < 1. Furthermore, for for all x is an element of <((Omega))over bar>, the function mu(x) lie in the psi(theta 1 theta 2), where psi(theta 1 theta 2) = {z is an element of C : pi/2 < theta(1) <= vertical bar arg z vertical bar <= theta(2) < pi}.