Sturm-Liouville systems on a class of Hilbert spaces

Anthony Hastir, Judicaël Mohet, Joseph J. Winkin

Published 2022-07-20Version 1

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 central result of this note. Properties related to the C_0-semigroup generated by the opposite of such Riesz-spectral operator are also highlighted. The main result is 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. The advantage of performing such an approach is illustrated on this example by means of an observability analysis.