The non-equivalence of pseudo-Hermiticity

and presence of antilinear symmetry

PETR SIEGL^1,2,3

¹Faculty of Nuclear Sciences and Physical Engineering, Prague, Czech Republic

²Department of Theoretical Physics, Nuclear Physics Institute,

Academy of Sciences, \v Re\v z, Czech Republic

³Laboratoire Astroparticule et Cosmologie, Universit\'e Paris 7, Paris, France

E-mail: siegl@ujf.cas.cz

Abstract. The non-equivalence of the presence of antilinear symmetry and

pseudo-Hermiticity is shown for bounded operators. Two appropriate

examples are operators with non-empty residual spectrum. The class

of operators for which the equivalence holds is extended to the

spectral operators of scalar type. The importance of

$J$-self-adjointness is stressed and new proofs using this property

are provided.

Keywords. Pseudo-Hermiticity; antilinear symmetry; $J$-self-adjoint

operators.

PACS Nos 03.65.Db; 02.30.Tb; 03.65.Ca