Decomposability of Krein space operators

Abstract

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 $(\beta)$ is decomposable, and study the relation between decomposability and property $(\beta)$ for $\mathcal{J}$-selfadjoint operators.

We prove the equivalence of these properties for $\mathcal{J}$-selfadjoint operators $T$ and $T^*$ by using their local spectra and local spectral subspaces.