Let \(\Pi\) be a Krein space and let U be a unitary operator in \(\Pi\). First the author considers the decomposition of \(\Pi\) into spectral subspaces of U corresponding to a decomposition of the spectrum of U into spectral sets [cf. e.g. H. Langer, Lect. Notes Math. 948, 1-46, § I.3 (1982; Zbl 0511.47023)]. Then some examples related to the spectrum of operators in Krein spaces are given: an invertible, bounded, quasinilpotent selfadjoint operator, a selfadjoint operator with empty spectrum, a selfadjoint operator A with non-densely defined \(A^ 2\) and others. Furthermore, the maximal nonnegative subspaces of (H,[A\(\cdot,\cdot])\), where (H,[\(\cdot,\cdot])\) is a Hilbert space and A is an invertible, bounded selfadjoint operator in (H,[\(\cdot,\cdot])\), are characterized in terms of their angular operators [cf. e.g. J. Bognár, Indefinite inner product spaces (1974; Zbl 0286.46028), Theorem II.11.7].