On Krein's formula in indefinite metric spaces

• Published in: Linear algebra and its applications Vol. 389; pp. 305 - 322
• Main Authors: Belyi, Sergey, Tsekanovskii, Eduard
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 15-09-2004; Elsevier Science
• Subjects: Algebra; Exact sciences and technology; Indefinite metrics; Krein's formula; Linear and multilinear algebra, matrix theory; Mathematics; Pontryagin space; Sciences and techniques of general use; Pontryagin space; Indefinite metrics; Krein's formula; Neumann problem; Resolvent; Metric space; Indefinite metric; Parameterization; Linear transformation
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2004.04.002

In this paper we extend some of the recent results in connection with the Krein resolvent formula which provides a complete description of all canonical resolvents and utilizes Weyl–Titchmarsh functions in the spaces with indefinite metrics. We show that coefficients in Krein's formula can be expressed in terms of analogues of the von Neumann parametrization formulas in the indefinite case. We consider properties of Weyl–Titchmarsh functions and show that two Weyl–Titchmarsh functions corresponding to
π-self-adjoint extensions of a densely defined
π-symmetric operator are connected via linear-fractional transformation with the coefficients presented in terms of von Neumann's parameters.