Operator (quasi-)similarity, quasi-Hermitian operators and all that
Abstract
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), in particular the scale of Hilbert spaces generated by a single unbounded metric operator.
Keywords
Cite
@article{arxiv.1510.02723,
title = {Operator (quasi-)similarity, quasi-Hermitian operators and all that},
author = {Jean-Pierre Antoine and Camillo Trapani},
journal= {arXiv preprint arXiv:1510.02723},
year = {2016}
}
Comments
16 pages. arXiv admin note: text overlap with arXiv:1409.3497, arXiv:1210.3163, arXiv:1307.5644