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

Abstract

In the present paper we investigate the set $\hat{I}{\pounds }_J$ 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 $\mathcal{C}{l}_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-RJ$. This enables one to construct the collection of operators $C_{\ddot{I}‡,\ddot{I}‰}$ realizing the property of stable $C$-symmetry for extensions $A\in \hat{I}{\pounds }_J$ directly in terms of $\mathcal{C}{l}_2(J,R)$ 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.