On a class of J-self-adjoint operators with empty resolvent set

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:379

Article

Kuzhel, Sergii¹  Trunk, Carsten² 

[1] AGH Univ Sci & Technol, Dept Appl Math, PL-30059 Krakow, Poland

[2] Tech Univ Ilmenau, Inst Math, D-98684 Ilmenau, Germany

关键词: Krein spaces;    J-self-adjoint operators;    Empty resolvent set;    Stable C-symmetry;    Sturm Liouville operators;   

DOI  :  10.1016/j.jmaa.2010.12.048

来源: Elsevier

【 摘 要 】

In the present paper we investigate the set Sigma(J) of all J-self-adjoint extensions of an operator S which is symmetric in a Hilbert space (sic) with deficiency indices (2, 2) and which commutes with a non-trivial fundamental symmetry J of a Krein space ((sic), [.,.]), SJ = JS. Our aim is to describe different types of J-self-adjoint extensions of S. which, in general, are non-self-adjoint operators in the Hilbert space (sic). One of our main results is the equivalence between the presence of J-self-adjoint extensions of S with empty resolvent set and the commutation of S with a Clifford algebra Cl-2(J, R), where R is an additional fundamental symmetry with JR = -RJ. This enables one to parameterize in terms of Cl-2(J, R) the set of all J-self-adjoint extensions of S with stable C-symmetry. Here an extension has stable C-symmetry if it commutes with a fundamental symmetry and, in turn, this fundamental symmetry commutes with S. Such a situation occurs naturally in many applications, here we discuss the case of indefinite Sturm-Liouville operators and the case of a one-dimensional Dirac operator with point interaction. (C) 2010 Elsevier Inc. All rights reserved.

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