Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces
(2015) Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects — pp. 345-402, published
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- Authors
Antoine, Jean-Pierre 
- Abstract
- A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.
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Citations
Antoine, J.-P., & Trapani, C. (2015). Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces. In F. Bagarello, J-P. Gazeau, F. H. Szafraniec and M. Znojil (eds.) (ed.), Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects (p. pp. 345-402). J.Wiley and Sons. https://doi.org/10.1002/9781118855300.ch7