Among various available approaches to deformations of Heisenberg
algebra the authors start from a q-deformed commutator between the
creation and annihilation operator [cf. their Eq.(6)]. They deduce
(firstly) the standard canonical commutation relations [between the
dynamical variables X and P -- typically, Eq. (10) or (15)] and
(secondly) their non-Hermitian realizations which they (thirdly)
Hermitize using the Dyson-mapping technique of Ref. [13b].

After some review text the main technical message comes with the
turn of attention to the manifestly non-Hermitian toy-model
Hamiltonian of Ref. [16] where one must apply still another, tilded
version $\tilde{\eta}$ of the Dyson map ${\eta}$. Luckily, the two
operators $\tilde{\eta}$ and  ${\eta}$ commute. Hence, in a climax
of their text the authors are able to compute the minimal
uncertainty and the minimal length pertaining to the model.

A comment on priorities deserves to be added. While the paper under
review (let us call it paper II) has been made publicly available on
July 30th, 2009 (via arXiv), another submission (let us call it
paper I, now published: T.K. Jana and P. Roy, Non-Hermitian Quantum
Mechanics with Minimal Length Uncertainty, SIGMA, Vol. 5 (2009),
083, 7 pages) on similar subject already resided safely in my own
Guest-editor's (plus in three pairs of the referees') hands. Still,
being submitted as early as on June 30th, and being send for a minor
revision on July 21st, paper I has only been made publicly
available, via arXiv, on August 12th. For this reason the authors of
paper II felt annoyed by not being cited (this is proved by their
Comment in SIGMA, Vol. 5 (2009), 089, 2 pages) while the authors of
paper I felt equally annoyed for not being given the space for their
Reply (found too delayed by the Office of SIGMA -- its unpublished
version is, fortunately, available via web: see arXiv:0910.5601).

Last but not least, who also felt annoyed was the present reviewer
because nobody cited his own, closely related paper ``Fundamental
length in quantum theories with PT-symmetric Hamiltonians" [Phys.
Rev. D 80, 045022 (2009), submitted on April 16th]. Fortunately for
all of us (and in a sharp contrast to the words ``... no motivation
is given for why one would want to consider non-hermitian
Hamiltonians - this is my main criticism" by one of anonymous
referees of paper I) the idea proved productive. One of its very
nice recent further developments has been described by A. Fring, L.
Gouba and F. G. Scholtz in paper ``Strings from dynamical
noncommutative space-time" (arXiv:1003.3025).

MR2552469 Bagchi, Bijan; Fring, Andreas Minimal length in quantum
mechanics and non-Hermitian Hamiltonian systems. Phys. Lett. A 373
(2009), no. 47, 4307--4310. 81Q12 (81S05)