The fine spectra of a difference of weighted shift operators acting in some classical sequence spaces

In this paper we investigate the fine spectra, the ergodic properties and the linear dynamics of the operator $C(x)=(−nx_n+n(n−1)x_{n−2})_{n∈N}, x=(x_n)_{n∈N}∈C^N$, and of its formal adjoint C′ acting in the classical sequence spaces s, s′, ω and φ. The operators C and C′ acting in s are conjugate to the one-dimensional Ornstein-Uhlenbeck differential operator Ou=u″−xu′ and to the differential operator $O′u=u″+xu′+u$ acting in $S(R)$, respectively.