The linear operator \(T\) in an inner product space \(({\mathcal K}, [\cdot, \cdot ])\) is called contractive (expansive, resp.) if \([Tx, Tx ]\leq [x,x ]\) (\([Tx, Tx]\geq [x,x ]\), resp.) for all \(x\in {\mathcal K}\). Eigenvalues, in particular those in the unit disc, and the signatures of the corresponding eigenspaces were studied e.g. in [\textit{I. S. Iokhvidov}, \textit{M. G. Krein} and \textit{H. Langer}, `Introduction to the spectral theory of operators in spaces with an indefinite metric' (1982; Zbl 0506.47022); \textit{T. Ya. Azizov} and \textit{I. S. Iokhvidov}, `Linear operators in spaces with an indefinite metric' (1989; Zbl 0714.47028) and \textit{L. de Branges}, Trans. Am. Math. Soc. 305, 277-291 (1988; Zbl 0647.46027)], where also references to earlier papers can be found. It is the aim of this note to prove results of this type under fairly general assumptions, to improve earlier results, e.g. Lemma 11.8 of the first named book about an expansive but not doubly expansive operator, and to show that e.g. in a Pontrjagin space the inner product on an eigenspace of a contraction \(T\) at \(z\), \(|z|=1\), is very similar to the inner product on an eigenspace at \(z\) of a unitary operator. Most of the statements below have analogues for operators which are dissipative with respect to an indefinite inner product, and, in fact, some of them were earlier proved in this context [see the second named book]. The formulation of these analogues is left to the reader.