Analysis of spectral points of the operators \(T^{[*]} T\) and \(TT ^{[*]}\) in a Krein space (Q1030504)

Let \(T\) be a (in general unbounded) linear operator in a Krein space and denote by \(T^+\) the adjoint with respect to the Krein space inner product. The main objective of the present paper is to study the connections between spectral points of the operators \(T^+ T\) and \(TT^+\). If \(T\) is bounded, then it is well-known that the nonzero spectra of these two operators coincide. In the unbounded case, not much can be said without additional assumptions. The principal assumption in this paper is that both operators are definitizable in the sense of \textit{H.\,Langer} [Lect.\ Notes Math.\ 948, 1--46 (1982; Zbl 0511.47023)]. Under this assumption, the extended real line divides into four parts with respect to both operators \(T^+ T\) and \(TT^+\): Resolvent set, spectral points of definite type, regular critical points and singular critical points. The points of special interest for \(T^+ T\) and \(TT^+\) are \(0\) and \(\infty\). For the point \(0\) it is shown, e.g., that \(0\) can not be in the resolvent set of \(T^+T\) and a singular critical point of \(TT^+\) at the same time, but all other combinations are possible.