The positive radial momentum operator

The Laplacian in spherical coordinates contains the squared radial momentum operator $p_r^2 = - r^{-1} \partial_r^2 r$ which is Hermitian and positive. However as has been pointed out by various authors the ``radial momentum operator" $- i r^{-1} \partial_r r$ is not Hermitian. The positive square root operator of $p_r^2$ is found and also its inverse. We discuss the relation of these operators with Fourier transforms, the Hilbert transform and fractional integral operators.