Locally Definitizable Operators: The Local Structure of the Spectrum

Locally definitizable operators are a subclass of self-adjoint operators in Krein spaces. They have locally the same spectral properties as definitizable operators in Krein spaces. It is shown in this note how to define spectral points of positive/negative type and spectral points of type π+∕π− via approximative eigensequences. This approach has the advantage that it does not make use of a local spectral function. Moreover, perturbation results for locally definitizable operators are discussed. Spectral points of type π+ and π− are stable under compact perturbations. For real spectral points of type π+ and type π−, which are not in the interior of the spectrum the growth of the resolvent in an open neighborhood of these spectral points is of finite order. This can be utilized to show the existence of a local spectral function with singularities. In turn, with the help of this local spectral function, one characterizes spectral points of positive/negative type and spectral points of type π+ and type π−: All spectral subspaces corresponding to sufficiently small neighborhoods of spectral points of positive/negative type are Hilbert or anti-Hilbert spaces, and all spectral subspaces corresponding to spectral points of type π+ or type π− are Pontryagin spaces. Locally definitizable operators are used in the study of indefinite Sturm–Liouville problems, λ-dependent boundary value problems, \({\mathcal P}{\mathcal T}\) -symmetric operators, partial differential equations and in the study of problems of Klein–Gordon type.