In this paper, we review some properties in the local spectral theory and
various subclasses of decomposable operators. We prove that every Krein
space selfadjoint operator having property (?) is decomposable, and clarify
the relation between decomposability and property (?) for J-selfadjoint
operators. We prove the equivalence of these properties for J-selfadjoint
operators T and T* by using their local spectra and local spectral
subspaces.
Decomposability of Krein space operators
