Spectral Doubling of Normal Operators and Connections with Antiunitary Operators

Equations such as AB = B ^T A have been studied in the finite dimensional setting in (Linear Algebra Appl 369:279–294, 2003). These equations have implications for the spectrum of B, when A is normal. Our aim is to generalize these results to an infinite dimensional setting. In this case it is natural to use JB*J for some conjugation operator J in place of B ^T . Our main result is a spectral pairing theorem for a bounded normal operator B which is applied to the study of the equation KB = B*K for K an antiunitary operator. In particular, using conjugation operators, we generalize the notion of Hamiltonian operator and skew-Hamiltonian operator in a natural way, derive some of their properties, and give a characterization of certain operators B for which AB = (JB*J)A and BA = A(JB*J) and also those B with KB = B*K for certain antiunitary operators K.