  In this paper we extend Sylvester's approach via upper triangular compact
operators to establish the discreteness of transmission eigenvalues for
higher-order main terms and higher-order perturbations. The coefficients of the
perturbations must be sufficiently smooth and the coefficients of the
higher-order terms of the perturbation must vanish in a neighbourhood of the
boundary of the underlying domain. The zeroeth order term must satisfy a
suitable coercivity condition in a neighbourhood of the boundary.
