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

Abstract

In the present paper we investigate the set ΣJ of all J-self-adjoint extensions of an operator S which is symmetric in a Hilbert space H with deficiency indices 〈2,2〉 and which commutes with a non-trivial fundamental symmetry J of a Krein space (H,[⋅,⋅]),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 H. 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 Cl2(J,R), where R is an additional fundamental symmetry with JR=−RJ. This enables one to parameterize in terms of Cl2(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.