Rigorous and extremely interesting contribution to the subject
explained by the title. As the fourth item in the author's 2010
series it extends some of his results to the case of operators which
are not self-adjoint in a {\em given}, ``false" Hilbert space.
Typically, one has in mind quantum Hamiltonians a.k.a.
``PT-symmetric" or ``cryptohermitian"; they ARE self-adjoint in
ANOTHER, ``standard" Hilbert space after all.

Although the author himself characterizes his effort as motivated by
the needs of quantum physics, the impact of his message in the
community might be weakened by his ``original" notation. This
creates an unnecessary obstacle. {\it Pars pro toto}, the
Hamiltonian operators are mostly found denoted by symbols
``$\Theta$" which, in turn, is precisely the letter used by some
other authors for his present $x$, etc. This being said, the need of
the rigorous treatment of the subject has been urgent. The more one
regrets that this ``applicability" point is not treated properly.
E.g., the lack of references to at least a few key papers in the
field would certainly amend its possible influence. One misses, most
badly, the reference to a topic which certainly is particularly
close to the author's range of tastes, viz., to the use of auxiliary
bosons in nuclear physics (cf. Scholtz et al, 1992).

The impact of the message in the mathematically oriented community
might equally suffer from the absence of a few other standard
references. {\it Pars pro toto} one misses the Krein-space
connection (Langer, Veselic, plus monographs). The
reference~[11]-[14] to the recent re-rediscoveries of the idea is
certainly insufficient. The readership would certainly profit also
from the citation of J. Dieudonn\'e's ``Quasi-Hermitian operators"
[Proc. Int. Symp. Lin. Spaces, Pergamon, Oxford, 1961, pp. 115-122]
and to some of its sequels where the subject is called,
incidentally, ``quasi-Hermiticity" and where the key message is not
too far from the present one (I have in mind, e.g., that ``it is not
necessary that $x^{-1}$ exists").