Product systems arising from Lévy processe

This paper investigates the structure of product systems of Hilbert spaces derived from Banach space-valued Lévy processes. We establish conditions under which these product systems are completely spatial and show that Gaussian Lévy processes with non-degenerate covariance always give rise to product systems of type I. Furthermore, we construct a continuum of non-isomorphic product systems of type $\mathrm{I}\mathrm{I}\text{sb}\mathrm{\infty }$ from pure jump Lévy processes.