In recent years, there has been a rapid growth of interest in spectral properties of non-self-adjoint operators and operator pencils. This thesis is concerned with indefinite self-adjoint linear pencils which lead to a special class
of non-self-adjoint spectral problems. These problems are not well understood,
and, in general, many sign-indefinite problems which are trivial to state require
some highly non-trivial analysis.
We look at indefinite linear pencil problems from the perspective of a two parameter eigenvalue problem. We derive localisation results for real eigenvalues and present several examples. We also use different approaches to obtain
estimates of non-real eigenvalues, supported by a large number of numerical
experiments. Additionally, these experiments lead to various open questions
and conjectures
