  The class of Sturm-Liouville operators on the space of square integrable
functions on a finite interval is considered. According to the Riesz-spectral
property, the self-adjointness and the positivity of such unbounded linear
operators on that space, a class of Hilbert spaces constructed as the domains
of the positive (in particular, fractional) powers of any Sturm-Liouville
operator is considered. On these spaces, it is shown that any Sturm-Liouville
operator is a Riesz-spectral operator that possesses the same eigenvalues as
the original ones, associated to rescaled eigenfunctions. This constitutes the
first central result of this paper. Properties related to the C_0-semigroup
generated by the opposite of such Riesz-spectral operator are also highlighted.
In addition as second central result, a characterization of approximate
observability by means of point measurement operators is established for such
systems. The main results are applied on a diffusion-convection-reaction system
in order notably to show that the dynamics operator is the infinitesimal
generator of a compact C_0-semigroup on some Sobolev space of integer order,
and to establish its observability.
