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