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Partially Fundamentally Reducible Operators in Krein Spaces
Integral Equations and Operator Theory (2015)
  • Branko Ćurgus, Western Washington University
  • Vladimir Derkach
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.
  • Self-adjoint extension,
  • Symmetric operator,
  • Kreĭn space,
  • Fundamentally reducible operator,
  • Coupling of operators,
  • Boundary triple,
  • Weyl function,
  • Similar to a self-adjoint operator
Publication Date
July, 2015
Publisher Statement
© Springer International Publishing AG, Part of Springer Science+Business Media DOI: 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)
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