Abstract
We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators.
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Acknowledgements
D.K. is grateful to the Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, where the ideas of this paper were discussed, for partially supporting his stays in 2020 and 2021.
Funding
C.C. was partially supported by FCT/Portugal through CAMGSD, IST-ID, projects number UIDB/04459/2020 and UIDP/04459/2020. D.K. was partially supported by the EXPRO Grant No. 20-17749X of the Czech Science Foundation.
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