  We review properties of q-orthogonal polynomials, related to their
orthogonality, duality and connection with the theory of symmetric
(self-adjoint) operators, represented by a Jacobi matrix. In particular, we
show how one can naturally interpret the duality of families of q-polynomials,
orthogonal on countable sets of points. In order to obtain orthogonality
relations for dual sets of polynomials, it is proposed to use two symmetric
(self-adjoint) operators, representable (in some distinct bases) by Jacobi
matrices. This approach is applied to several pairs of dual families of
q-polynomials, orthogonal on countable sets, from the q-Askey scheme. For each
such pair, the corresponding operators, representable by Jacobi matrices, are
explicitly given. These operators are employed in order to find explicitly
sets, on which the polynomials are orthogonal, and orthogonality relations for
them.
