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

On explicit realization of algebra of complex powers of generators of $U_{q}(\mathfrak{sl}(3))$