Article

Partially Fundamentally Reducible Operators in Krein Spaces

Integral Equations and Operator Theory
(2015)
Abstract

A self-adjoint operator

*A*in a Kreĭn space (K,[⋅,⋅]) is called partially fundamentally reducible if there exist a fundamental decomposition K=K+[+˙]K− (which does not reduce*A*) and densely defined symmetric operators S + and S −in the Hilbert spaces (K+,[⋅,⋅]) and (K−,−[⋅,⋅]), respectively, such that each S + and S − has defect numbers (1, 1) and the operator A is a self-adjoint extension of S=S+⊕(−S−) in the Kreĭn space (K,[⋅,⋅]). The operator*A*is interpreted as a coupling of operators S + and −S−relative to some boundary triples (C,Γ+0,Γ+1) and (C,Γ−0,Γ−1). Sufficient conditions for a nonnegative partially fundamentally reducible operator*A*to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions m + and m − of S + and S − relative to the boundary triples (C,Γ+0,Γ+1) and (C,Γ−0,Γ−1). Moreover, it is shown that under some asymptotic assumptions on m + and m − all positive self-adjoint extensions of the operator S are similar to self-adjoint operators in a Hilbert space.Keywords

- Self-adjoint extension,
- Symmetric operator,
- Kreĭn space,
- Fundamentally reducible operator,
- Coupling of operators,
- Boundary triple,
- Weyl function,
- Similar to a self-adjoint operator

Disciplines

Publication Date

July, 2015
Publisher Statement

© Springer International Publishing AG, Part of Springer Science+Business Media DOI: 10.1007/s00020-014-2204-3 http://link.springer.com/article/10.1007/s00020-014-2204-3

Citation Information

Branko Ćurgus and Vladimir Derkach. "Partially Fundamentally Reducible Operators in Krein Spaces" *Integral Equations and Operator Theory*Vol. 82 Iss. 4 (2015)

Available at: http://works.bepress.com/branko_curgus/21/