Yes, diffusion operators in dimensions [$]d \ge 2[$] are generally not self-adjoint, regardless of the boundary conditions or linear space.  

However, there are analogs of the Perron-Frobenius theorem for matrices that can apply to those elliptic operators, and that may be relevant to what you are trying to argue. For example, Krein-Rutman here:
ftp://ftp.ma.utexas.edu/pub/papers/llav ... hap1-1.pdf

When the [$]d \ge 2[$] diffusion operators [$]A[$] are not self-adjoint, but still have a discrete spectrum, there is typically a bi-orthonormal eigenfunction expansion for the solutions to evolution problems [$]f_t = A f[$]. The expansion uses eigenvalues [$]\{\lambda_n\}[$] that generally come in complex-conjugate pairs. But the principal eigenvalue [$]\lambda_0[$] is still real (by the generalized Perron-Frobenius stuff), it comes with a positive eigenfunction [$]u_0[$],and it's the [$](\lambda_0,u_0)[$] pair that characterizes the [$]T \rightarrow \infty[$] behavior.