Local Definitizability of T^([*])T and TT^([*])

The spectral properties of two products $AB$ and $BA$ of possibly unbounded operators $A$ and $B$ in a Banach space are considered. The results are applied in the comparison of local spectral properties of the operators $T^{[*]} T$ and $TT^{[*]} $ in a Krein space. It is shown that under the assumption that both operators $T^{[*]} T$ and $T^{[*]} $ have non-empty resolvent sets, the operator $T^{[*]} T$ is locally definitizable if and only if $TT^{[*]} $ is. In this context the critical points of both operators are compared.