  We provide the mathematical foundation for the $X_m$-Jacobi spectral theory.
Namely, we present a self-adjoint operator associated to the differential
expression with the exceptional $X_m$-Jacobi orthogonal polynomials as
eigenfunctions. This proves that those polynomials are indeed eigenfunctions of
the self-adjoint operator (rather than just formal eigenfunctions). Further, we
prove the completeness of the exceptional $X_m$-Jacobi orthogonal polynomials
(of degrees $m, m+1, m+2, ...$) in the Lebesgue--Hilbert space with the
appropriate weight. In particular, the self-adjoint operator has no other
spectrum.
