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.

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