We continue the work of [Camano, Lackner, Monk, SIAM J.\ Math.\ Anal., Vol.\ 49, No.\ 6, pp.\ 4376-4401 (2017)]
on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues due not correspond
to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties.
The present article considers the original and the modified electromagnetic Steklov eigenvalue problem.
We cast the problems as eigenvalue problem for a holomorphic operator function $A(\cdot)$. We construct a
``test function operator function'' $T(\cdot)$ so that $A(\lambda)$ is weakly $T(\lambda)$-coercive for all
suitable $\lambda$, i.e.\ $T(\lambda)^*A(\lambda)$ is a compact perturbation of a coercive operator.
The construction of $T(\cdot)$ relies on a suitable decomposition of the function space into subspaces and an apt
sign change on each subspace.
For the approximation analysis, we apply the framework of T-compatible Galerkin approximations.
For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence
convergence.
For the original problem, we require the projection operators to satisfy an additional commutator property involving
the tangential trace. The existence and construction of such projection operators remain open questions.
A Holomorphic Operator Function Approach for the Laplace Eigenvalue Problem Using Discontinuous Galerkin Method
