Some Questions in the Perturbation Theory of J-Nonnegative Operators in Krein Spaces

✍ Scribed by P. Jonas; H. Langer

• Publisher: John Wiley and Sons
• Year: 1983
• Tongue: English
• Weight: 958 KB
• Volume: 114
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19831140116

✦ Synopsis

In this note we study perturbations of a J-nonnegative operator A in a KREIN space which are such that the difference of the resolvents of A and of the perturbed operator B is of rank one. Here B is also supposed to be J-selfadjoint. With the pair A, B we associate a one-parameter family {Br),,eR of J-selfadjoint operators (or linear relations) which extend the "common part" of A and B. In particular, A and B belong to this family. Furthermore, we characterize those opemtors in the family which are J-nonnegative (Theorem 2.3.2). The description of the operators in this family through the corresponding resolvents is similar to M. G. KREIN'S description of the orthogonal generalized resolvents of a EhsmTian operator with defect numbers one in HILBHRT space (see e.g. [l]). The Theorem 2.3.2 gives, in particular, a description of the generalized resolvents of a densely defined J-nonnegative HrPmmian operator A, with defect one on some KBEIN space S, which are generated by J-nonnegative J-selfadjoint extensions of A, in S.