Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators

We show that the interior transmission eigenvalues are discrete by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary.