Analysis of spectral points of the operators \(T^{[*]} T\) and \(TT ^{[*]}\) in a Krein space
type: Item
instance of: scholarly article
title: Analysis of spectral points of the operators \(T^{[*]} T\) and \(TT ^{[*]}\) in a Krein space
zbMATH Open document ID: 1189.47037
author: Michał Wojtylak
author: André C. M. Ran
published in: Integral Equations and Operator Theory
publication date: 2009-07-01T00:00:00Z
review text: Let \(T\) be a (in general unbounded) linear operator in a Krein space and denote by \(T^+\) the adjoint with respect to the Krein space inner product. The main objective of the present paper is to study the connections between spectral points of the operators \(T^+ T\) and \(TT^+\). If \(T\) is bounded, then it is well-known that the nonzero spectra of these two operators coincide. In the unbounded case, not much can be said without additional assumptions. The principal assumption in this paper is that both operators are definitizable in the sense of \textit{H.\,Langer} [Lect.\ Notes Math.\ 948, 1--46 (1982; Zbl 0511.47023)]. Under this assumption, the extended real line divides into four parts with respect to both operators \(T^+ T\) and \(TT^+\): Resolvent set, spectral points of definite type, regular critical points and singular critical points. The points of special interest for \(T^+ T\) and \(TT^+\) are \(0\) and \(\infty\). For the point \(0\) it is shown, e.g., that \(0\) can not be in the resolvent set of \(T^+T\) and a singular critical point of \(TT^+\) at the same time, but all other combinations are possible.
reviewed by: Jussi Behrndt
Mathematics Subject Classification ID: 47B50
zbMATH DE Number: 5573993
zbMATH Keywords: regular critical point
zbMATH Keywords: singular critical point
zbMATH Keywords: Jordan chain
zbMATH Keywords: Kreĭn space
MaRDI profile type: Publication
full work available at URL: s00020-009-1654-5
OpenAlex ID: W1969365709
DOI: 10.1007/S00020-009-1654-5
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description: scientific article; zbMATH DE number 5573993