  An open U_q(sl_2)-invariant spin chain of spin S and length N with
inhomogeneous coupling is investigated as an example of a non-Hermitian
(quasi-Hermitian) model. For several particular cases of such a chain, the
ranges of the deformation parameter gamma are determined for which the spectrum
of the model is real. For a certain range of gamma, a universal metric operator
is constructed and thus the quasi-Hermiticity of the model is established. The
constructed metric operator is non-dynamical, its structure is determined only
by the symmetry of the model. The results apply, in particular, to all known
homogeneous U_q(sl_2)-invariant integrable spin chains with nearest-neighbour
interaction. In addition, the most general form of a metric operator for a
quasi-Hermitian operator in finite dimensional space is discussed.
