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

Abstract

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