  We study a one-parameter family of self-adjoint normal operators for the
X-ray transform on the closed Euclidean disk ${\mathbb D}$, obtained by
considering specific singularly weighted $L^2$ topologies. We first recover the
well-known Singular Value Decompositions in terms of orthogonal disk (or
generalized Zernike) polynomials, then prove that each such realization is an
isomorphism of $C^\infty({\mathbb D})$. As corollaries: we give some range
characterizations; we show how such choices of normal operators can be
expressed as functions of two distinguished differential operators. We also
show that the isomorphism property also holds on a class of constant-curvature,
circularly symmetric simple surfaces. These results allow to design functional
contexts where normal operators built out of the X-ray transform are provably
invertible, in Fr\'echet and Hilbert spaces encoding specific boundary
behavior.
