Following the pioneering review paper by F.G. Scholtz, H.B. Geyer
and F.J.W. Hahne [Ann. Phys. (N.Y.) 213 (1992), p. 74], the
practical (e.g., variational) tractability of a broad family of
phenomenological models in quantum mechanics can be enhanced by
the use of certain non-Hermitian operator representations H of
observables which only become Hermitian with respect to a specific
and nonstandard ad hoc scalar product. Recently, a special
implementation of the same idea has been proposed by Bender et al.
(cf., e.g., the compact review by C.M. Bender [Czech. J. Phys. 54
(2004), p. 1027]) who postulated, in addition, the auxiliary
relation H PT = PT H where, originally, the Hermitian operator P
of parity was multiplied by T representing time reversal. This
opened the scene for the short Mostafazadeh's paper in question.
The rest of the story is summarized there and it is shown that and
how one can relate the notions of the PT symmetry based on
Hermitian Ps (i.e., in the author's language, the
P-pseudo-Hermiticity) and of the PT symmetry based on
non-Hermitian Ps (i.e., in the author's language, the weak
P-pseudo-Hermiticity). The text is compact, its definitions are
clear and well formulated, its propositions and a theorem are
carefully and responsibly proved. A nice and rewarding
mathematical reading, especially for the author of the first,
three-dimensional matrix example in section III (cf. ref. [7], to
be complemented here, perhaps, by its longer continuation with
dimensions running up to seven: Miloslav Znojil, Exactly solvable
models with PT-symmetry and with an asymmetric coupling of
channels, J. Phys. A: Math. Gen. 39 (2006) 4047 - 4061).


MR2262672 Mostafazadeh, Ali Is weak pseudo-Hermiticity weaker than
pseudo-Hermiticity? J. Math. Phys. 47 (2006), no. 9, 092101, 7 pp.
81Q10 (47N50)