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