On the regularity of the critical point infinity of definitizable operators. (English) Zbl 0572.47023

For a definitizable operator A in a Krein space \({\mathcal K}\) [for the terminology and fundamental results on this topic see: H. Langer, Proc. Conf., Dubrovnik, Yugoslavia 1981, Lect. Notes Math. 948, 1-46 (1982; Zbl 0511.47023)] several criteria for the regularity of the critical point (c.p.) \(\infty\) are given. In particular it is shown that \(\infty\) is not a singular c.p. of A if and only if in the Krein space \({\mathcal K}\) there exists a positive, bounded and boundedly invertible operator W such that W\({\mathcal D}(A)\subseteq {\mathcal D}(A)\) (or W\({\mathcal D}[JA]\subseteq {\mathcal D}[JA]\), where \({\mathcal D}[JA]={\mathcal D}(| JA|^{1/2})\) and J is a fundamental symmetry on \({\mathcal K})\). These criteria are used to prove that the regularity of the c.p. \(\infty\) is preserved under some additive perturbations, as well as for certain operators which are related to A. Applications to indefinite Sturm- Liouville problems and connections with the results of R. Beals [J. Differ. Equations 56, 391-407 (1985; Zbl 0512.34017)] are indicated.


47B50 Linear operators on spaces with an indefinite metric
34L99 Ordinary differential operators
Full Text: DOI


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