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

In the present paper we investigate the set Σ_J of all J-self-adjoint extensions of a symmetric operator S with deficiency indices <2,2> which commutes with a non-trivial fundamental symmetry J of a Krein space \(mathfrak{H}, \[cdot,cdot\]\), SJ=JS. Our aim is to describe different types of J-self-adjoint extensions of S. 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 {mathcal C}l₂(J,R), where R is an additional fundamental symmetry with JR=-RJ. This enables one to construct the collection of operators C_χ,ω realizing the property of stable C-symmetry for extensions AinΣ_J directly in terms of {mathcal C}l₂(J,R) and to parameterize the corresponding subset of extensions with stable C-symmetry. Such a situation occurs naturally in many applications, here we discuss the case of an indefinite Sturm-Liouville operator on the real line and a one dimensional Dirac operator with point interaction.