On the negative squares of a class of self‐adjoint extensions in Krein spaces

Abstract

AbstractA description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function τ (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {D}_\kappa ^{n\times n}$\end{document}‐functions is introduced and, in particular, the properties of the inverse of a certain \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {D}_\kappa ^{2\times 2}$\end{document}‐function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non‐positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.