Operator (Quasi-)Similarity, Quasi-Hermitian Operators and All that

  • Jean-Pierre AntoineEmail author
  • Camillo Trapani
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)


Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties. Then we study quasi-Hermitian operators, bounded or not, that is, operators that are quasi-similar to their adjoint and we discuss their application in pseudo-Hermitian quantum mechanics. Finally, we extend the analysis to operators in a partial inner product space (pip-space), in particular the scale of Hilbert space s generated by a single unbounded metric operator.


Hilbert Space Unitary Operator Symmetric Operator Real Spectrum Graph Norm 
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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut de Recherche en Mathématique et PhysiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Dipartimento di Matematica e InformaticaUniversità di PalermoPalermoItaly

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