Spectral analysis of bounded operators on semi-Hilbertian spaces

Let H be a Hilbert space and let A be a positive bounded operator on H. An operator \(T \in B(H)\) is said to be A-invertible if there exists \(S \in B(H)\) such that \(ATS=AST=A.\) In this paper we develop spectral analysis in relation with this new notion of invertibility. We show that this concept is well compatible with the semi-Hilbertian structure of H generated by the “indefinite” metric operator A. As well as placing the classical notion of an invertible operator in an appropriate setting, this new notion seems to be interesting for studies in the framework of non-Hermitian quantum mechanics.