Local Definitizability of {{T^{[\ast]}T}} and {{TT^{{[\ast]}}}}

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 ${T^{\ast]} T}$ and ${TT^{\ast]}}$ in a Krein space. It is shown that under the assumption that both operators ${T^{\ast]} T}$ and ${TT^{\ast]}}$ have non-empty resolvent sets, the operator ${T^{\ast]} T}$ is locally definitizable if and only if ${TT^{\ast]}}$ is. In this context the critical points of both operators are compared.