$\mathrm{Q̂}$ operator for canonical quantum gravity

We study the properties of the $\mathrm{Q̂}\left[\omega \right]$ operator on the kinematical Hilbert space $\mathcal{H}$ for canonical quantum gravity. Its complete spectrum with respect to the spin network basis is obtained. It turns out that $\mathrm{Q̂}\left[\omega \right]$ is diagonalized in this basis, and it is a well defined self-adjoint operator on $\mathcal{H}.$ The same conclusions are also tenable on the $\mathrm{SU}\left(2\right)\mathrm{}$ gauge invariant Hilbert space with the gauge invariant spin network basis.