  In this note we prove an integral identity involving complex powers of
generators of quantum group $U_{q}(\mathfrak{sl}(3))$ considered as certain
positive operators in the setting of positive principal series representations.
This identity represents a continuous analog of one of the Lusztig's relations
between divided powers of generators of quantum groups, which play an important
role in the study of irreducible modules \cite{Lu 1}. We also give definitions
of arbitrary functions of $U_{q}(\mathfrak{sl}(3))$ generators and give another
proofs for some of the known results concerning positive principal series
representations of $U_{q}(\mathfrak{sl}(3))$.
