  We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla
u)]$, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric
tensor. Our main result shows that this spectrum can be derived from the
spectral decomposition $K=Q \Lambda Q^T$, where $Q=Q(x,y)$ is an orthogonal
matrix and $\Lambda=\Lambda(x,y)$ is a diagonal matrix. More precisely,
provided that $K$ is continuous, the spectrum equals the convex hull of the
ranges of the diagonal function entries of $\Lambda$. The involved domain is
assumed to be bounded and Lipschitz, and both homogeneous Dirichlet and
homogeneous Neumann boundary conditions are considered. We study operators
defined on infinite dimensional Sobolev spaces. Our theoretical investigations
are illuminated by numerical experiments, using discretized problems.
  The results presented in this paper extend previous analyses which have
addressed elliptic differential operators with scalar coefficient functions.
Our investigation is motivated by both preconditioning issues (efficient
numerical computations) and the need to further develop the spectral theory of
second order PDEs (core analysis).
