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We study some properties of the Rovelli-Smolin-DePietri volume operator inloop quantum gravity, which significantly simplify the diagonalization problemand shed some light on the pattern of degeneracy of the eigenstates. Theoperator is defined by its action in the spaces of tensor products$\mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_N}$ of the irreducibleSU(2) representation spaces $\mathcal{H}_{j_i}, i=1,\ldots,N$, labelled withspins $j_i\in \frac{1}{2}\mathbb{N}$. We restrict to the spaces of SU(2)invariant tensors (intertwiners) with all spins equal $j_1=\ldots=j_N=j$. Wecall them spin $j$ monochromatic intertwiners. Such spaces are important in thestudy of SU(2) gauge invariant states that are isotropic and can be applied toextract the cosmological sector of the theory. In the case of spin $1/2$ wesolve the eigenvalue problem completely: we show that the volume operator isproportional to identity and calculate the proportionality factor.

Properties of the Rovelli-Smolin-DePietri volume operator in the spaces of monochromatic intertwiners